Download 6. Pulse Amplification.

6. Pulse Amplification.
6.1. Introduction.
Output pulse energies from femtosecond lasers typically do not exceed a few
nanojouls, and peak powers of megawatts. For many applications, higher energies or peak
powers are required. Researchers are seeking methods to shorten pulses, to increase peak
powers and peak intensities on targets. Given the limits on further trimming of the pulse
duration, further increases in peak power and peak intensities can only be obtained by
increasing output energy. Amplification of energy from the femtosecond lasers makes
possible terawatt peak powers. Amplification, combined together with the initial pulse
stretching and compression as a final stage can convert terawatt systems into petawatt lasers
with subpicosecond pulses.
So far the highest energies, peak powers and irradiance can be achieved in Nd:glass
amplifiers, not those based on Ti:sapphire. The most powerful laser in the world (in 2003) is
“Vulcan” in Rutherford Appleton Laboratory, United Kingdom delivering 2.5 kJ in two 150
nm beams, 1 pW, 1021 W/cm2 and Nova system at Lawrence Livermore National Laboratory
delivering 1.3 kJ pulse at 800 ps that can be compressed to 430 fs to achieve 1.3 pW and
1021W/cm2.
When a laser pulse passes through an optically active material in which the population
inversion is maintained by a pumping source, it gains energy from the stimulated emission
generated by itself in the medium. As a result the output pulse is amplified (fig. 6.1).
pumping
input
pulse
active
medium
Fig. 6.1 Illustration of amplification
181
amplified
pulse
6.2. Theoretical Background.
Let us consider a mechanism of amplification in the three-level system (Fig. 6.2)
3
τ 32
Wp
τ 21
2 (m)
laser
transition
1 (n)
Fig. 6.2 Three-level system.
Let assume that the relaxation time τ32 for the 3 → 2 transition is short in comparison
with the lifetime of the E2 state τ21, which is a good approximation in solid-state lasers. This
denotes that the number of molecules N3 occupying the level E3 is negligible compared with
the number of molecules in N1 and N2 (N3 ≈ 0, and N1 + N2 = N0) because the molecules are
pumped almost immediately into the metastable level E2 with only a momentary stay in E3.
Based on this assumption, the change in the population is
N
dN 2
= W ( N1 − N 2 ) − 2 + W p N1 ,
τ 21
dt
(6.1)
dN 1
dN 2
=−
dt
dt
(6.2)
W = B21 ρ .
(6.3)
where
Neglecting spontaneous emission and pumping Wp during the pulse duration, which is usually
much shorter, the eq. (6.1) can be written in the form
dn
= −γWn ,
dt
(6.4)
n = N1 − N 2
(6.5)
where
182
denotes the population inversion, γ = 1 +
g2
(for three-level system, where g2, g1 are the
g1
degeneration numbers). One can show 6.1 that
W =
σI
hν
(6.6)
 J 
,ν
where σ [cm2] is the stimulated emission cross section, I is the radiation intensity 
2
 s cm 
is the frequency of the 2 → 1 transition. Substituting (6.6) into (6.4) and expressing the
intensity in terms of the photon density φ [photons/cm3]
I
=φ,
(6.7)
dn
= −γ cσ nφ .
dt
(6.8)
hν c
one gets
The rate at which the photon density changes in a small volume at x of an active medium is
equal to
∂φ
∂φ
= Wn −
c
∂t
∂x
(6.9)
where the first term describes the number of photons generated by the stimulated emission
and the second term – the flux of photons which flows out from that region.
Using once again (6.6) and (6.7) in (6.9) one gets
∂φ
∂φ
= cnσφ −
c.
∂t
∂x
(6.10)
Using the differential equations (6.8) and (6.10) one can solve them for various types of the
input pulse shapes 6.2-6.3. For a square pulse of duration tp and the initial photon density φ0 one
obtains:
−1

φ ( x, t ) 
x  

= 1 − [1 − exp(− σ n x )]exp − γσφ0  c (t −   .
φ0
c  



(6.11)
After passing through an active medium of length x = L the amplification is given by
∞
G=
∫ φ (L,t ) dt
−∞
φ0 t p
.
(6.12)
After substituting (6.11) into (6.12) and integrating one gets
G=
1
c γ σ φ0 t p
{
}
ln 1 + [exp (γ σ φ 0 τ 0 c ) − 1] e n σ L .
183
(6.13)
The equation (6.13) can be cast in a different form using the input energy Ein
Ein = c φ0 t p hν ,
(6.14)
and the saturation fluence that we defined in chapter 1 (eq. 1.39)
Es =
hν
γσ
=
E st
,
γ g0
(6.15)
where Est = hνn is the energy stored per volume, g0 = nσ is the small signal gain coefficient
(see chapter 1, eq. 1.23). Using the expressions for Ein and Es in (6.13) one obtains the energy
gain G
G=
E s    Ein
ln 1 + exp
Ein    E s
  
 − 1G0  ,
  
(6.16)
where G0 = exp(g0L) is the small-signal single-pass gain.
Consider limit cases:
1. The input signal is low, Ein E s << 1 . The eq. (6.16) can be approximated to the
exponential dependence on the active medium length L
G ≅ G0 = exp( g 0 L ) .
(6.17)
2. The high input signal, Ein E s >> 1 . Then, eq. (6.16) becomes
G ≅ 1+
Es
g0 L
Ein
(6.18)
with the linear dependence of the energy gain.
6.3. Design Features of Amplifiers.
For many scientific and practical applications we need powerful pulses
6.4-6.6
. The
pulses emitted from the oscillator can be amplified in devices called amplifiers. No single
amplifier configuration has emerged because one configuration can be more suitable than
other for a given application. Usually, the gain G which can be achieved and the energy which
can be extracted from the amplifier is the main parameter in the design of amplifiers.
There are four most important configurations for laser light amplification:
1. multistage power amplifier (fig. 6.3),
2. multipass amplifier (MPA) (fig. 6.4),
3. regenerative amplifier (RGA) (fig. 6.5),
4. chirped pulse amplifier (CPA) (fig. 6.6).
184
oscillator
rear
mirror
output
mirror
active
medium
laser amplifier
gain
medium
output
pumping
output
pumping
pump
trigger
pump
trigger
firing
signal
delay
Fig. 6.3 Oscillator-amplifier configuration
The multistage power amplifier consists of a laser oscillator and a pumped active
medium called amplifier. The amplifier is driven by the oscillator, which generates an initial
pulse of moderate power and energy. The pulse from the oscillator passes the active medium
of the amplifier and its power grows, in extreme cases, up to 100 times. The pulse can be
amplified further by adding the next amplifier in the configuration presented in fig. 6.3. These
configurations are used for longer pulses. The double-pass amplifiers are used for typical
picosecond system and can deliver 125 mJ pulses of less than 60 fs at 20 Hz with a solid-state
saturable-absorber-based oscillator.
pump pulse
ga
me in
diu
m
se
pul
t
u
inp
output pu
ls
e
Fig. 6.4 Multipass amplifier
Multipass amplifiers (fig. 6.4) are used when extremely short pulses are required,
shorter than 35 fs at 1 kHz and energy of 1.5 mJ. This configuration is simple and less
185
sensitive to thermal lensing, therefore provides excellent beam quality, and is relatively easy
u
outp
t pul
inpu
t pul
se
to calculate and compensate dispersion in the system.
se
λ/4
PC1
A
M1
Pockels
cell
gain
medium
Ti:sapphire
PC2
P
thin
layer
polarizer
Pockels
cell
M2
pump pulse
Second Harmonic Generation
(Nd : YLF, Q-switch)
Fig. 6.5 Regenerative amplifier
Regenerative amplifiers (fig. 6.5) are generally employed for ultrafast system with
relatively low output energies from femtosecond oscillators (a few nJ) to achieve the output
energy from the amplifier of around 1 mJ at 1 kHz. Higher energy can be obtained from
combination RGA/MPA methods that can deliver energy greater than 3.5 mJ per pulse and
higher at repetition rates of 1 kHz. Regenerative amplifiers are also used to produce powerful
picosecond pulses from a train of low-energy pulses of modelocked lasers. The regenerative
amplifiers have originally been pumped by flash lamps or by cw arc lamps
6.7-6.12
. More
recently, the Nd:YAG, Nd:YLF or Nd:glass amplifiers are pumped by diode arrays 6.13-6.17.
186
grating
grating
Stretcher
seed pulse
from oscillator
Compresor
grating
grating
outpu
λ/4
PC1
A
Pockels
cell
M1
t puls
e
PC2
gain
medium
Ti:sapphire
thin
layer
polarizer
Pockels
cell
M2
pump pulse
Fig. 6.6 Chirped pulse amplifier
Amplification of ultrashort pulses leads to enormously large peak powers that may be
far above the damage threshold of the active medium employed in the amplifier. To avoid this
problem the chirped-pulse amplification (fig. 6.6) was developed.
6.4. Regenerative Amplifier.
Now we will explain how the regenerative amplifier operates. One can see from fig.
6.5 that the typical commercially available regenerative amplifier consists of an active
medium (Ti:sapphire crystal), two Pockels cells (PC1 and PC2), λ/4 plate and thin-film
polarizer placed between two mirrors M1 and M2.
The gain medium is pumped with the second harmonic of the solid-state laser
Nd:YLF, Q-switched with pulses durations of 250 ns and the average energy of 8 W at the
Brewster’s angle.
The regenerative amplifier selects an individual pulse from a train of modelocked
pulses (called a seed pulse). The seed pulse is trapped in an amplifier where it is amplified at
each pass in the crystal. The pulse passes many times (usually about 10-20 times) through the
gain medium to get more energy. Once its energy increases by as much as 106, the amplified
pulse is removed from the amplifier as the output pulse that can be used for further
applications.
187
The number of passages in the regenerative amplifier depends on the round-trip time
between the M1 and M2 mirrors in the amplifier. If the round-trip time is 10 ns and the
pumping pulse duration is 250 ns, a typical time evolution of the pulse energy in a
regenerative amplifier is given in fig. 6.7.
output
energy
1
100
200
300
t [ns]
10
20
30
number
of steps
Fig. 6.7 Time evolution of the amplified pulse energy versus time and the number of
passages in the regenerative amplifier
The amplified pulse shape follows the shape of the pumping pulse originating from the
depletion of the excited-state in the gain medium. It is obvious from fig. 6.7 that the pulse
should be switched out from the amplifier after around 20 passages where there is a gain
maximum and the energy of the amplified pulse is the highest.
Now we have to understand how the input pulse is trapped in the amplifier. We will
show that it is trapped on the polarization basis. Let us recall that:
1. λ/4 plate changes the polarization from linear to circular,
2. λ/2 plate rotates the linear polarization by 2α, where α is the angle between the
polarization vector and the optic axis,
3. thin layer polarizer acts as a reflective polarizer (see chapter 1) that at the Brewster’s
angle reflects the polarization perpendicular to the plane of reflection, and transmits
the polarization parallel and perpendicular to the plane of reflection,
4. Pockels cell acts like a λ/4 or λ/2 plate depending on the external voltage applied..
188
6.4.1. Pockels Cell.
The Pockels cell plays an important role as an electro-optic Q-switch 6.18. Now we will
explain its operation. The Pockels cell consists of a nonlinear optical material with the voltage
applied to the material. The electric field can be applied along the direction of the optical
beam (fig. 6.8a) or perpendicular to it (fig. 6.8b).
a)
b)
z
nonlinear
material
(KD*P)
y
laser
beam
x
nonlinear
material
z
optic
axis
E
V
E
V
laser
beam
Fig. 6.8 The Pockels cell, the electric field can be applied along the direction of the optical
beam (a) or perpendicular to it (b)
The crystal becomes birefringent under the influence of the applied electric field. The crystals
used for parallel configuration (fig. 6.8a) are uniaxial in the absence of an electric field with
the optic axis along z direction. The ellipsoid of the refraction index projects as a circle on a
plane perpendicular to the optic axis (fig. 6.9a).
189
a)
b)
y’
y
y
z
optic
axis
x
z
optic
axis
x
x’
Fig. 6.9 Change of the refractive index in a KD*P crystal, x, y, z – the crystalographic axes
a) without an electric field, b) when an electric field is applied (E ≠ 0), x’, y’, z – the
electrically induced axes
So, the laser beams having polarizations along x or y directions propagate with the same
velocities along the z axis because the crystal is not birefringent in the direction of the optic
axis. For the situation presented in fig. 6.8a, the laser beam linearly polarized along y passes
through the KD*P crystal unchanged when no voltage is applied. When an electric field is
applied parallel to the crystal optic axis z, the ellipsoid of the refraction index projects as an
ellipse, not a circle, on the plane perpendicular to the optic axis with the axes x’ and y’
rotating by 45° with respect to the x and y crystallographic axes (fig. 6.9b). The angle of 45°
is independent of the magnitude of the electric field. Therefore, when a voltage is applied, the
KD*P crystal becomes birefringent along the z axis, and divides the laser beam into two
components (along x’ and y’) that travel through the crystal at different velocities. The
polarization of the output beam dependents on the phase difference between the two
ortogonally polarized components – ordinary and extraordinary rays, which depends on the
applied voltage
δ=
2π
λ
l∆n ,
(6.19)
where ∆n is the difference in the indices of refraction for the ordinary and extraordinary
beams, l is the crystal length.
It has been shown 6.19 that ∆n can be expressed as
∆n = n03 r63 E z ,
(6.20)
where r63 is the element of the electro-optic tensor of the third rank that is the response to the
applied field E in the z direction (Ez), n0 is the index of refraction for the ordinary ray.
190
Employing the relation between the voltage V and the applied electric field Ez, V z = E z l and
inserting (6.20) in eq. (6.19) one gets
δ=
2π
λ
n03 r63V z .
(6.21)
When the applied voltage Vz is adjusted to generate the phase difference δ = π/4 or δ = π/2 the
Pockels cell operates as a quarter-wave or half-wave plate.
The Pockels cell belongs to the fastest all-optical switching devices, and is highly
reliable. Typical commercially available Pockels cells employ KD*P crystals with λ/4 voltage
between 3.5 and 4 kV at 0.69 µm and 5 to 6 kV at 1.06 µm
6.1
. As a particular example,
Pockels cells are used to select and retain high-peak-power pulses in regenerative amplifiers.
Knowing how the Pockels cell operates let us return to fig. 6.5 to explain the
mechanisms of trapping the seed pulse from a train of modelocked pulses inside the amplifier.
The operation of a regenerative amplifier presented in fig. 6.5 can be divided into the
following steps:
1. A seed pulse arrives in the amplifier at the Brewster’s angle by reflecting from the
surface of the active medium (Ti+3:Al2O3). The polarization of the seed pulse is
perpendicular to the plane of drawing. The polarization of the pulse passing through the
Pockels cell PC1, which is initially switched off is unchanged, but double passing
through λ/4 (before and after reflection from M1) changes the polarization by 90°. So,
the beam reflected from M1 is not reflected from the crystal surface at A point but is
refracted and passes the crystal leading to its amplification by the stimulated emission
induced in the crystal (Ti+3:Al2O3) in which the population inversion is generated by the
pumping source. The pulse passes through the thin-layer polarizer and the Pockels cell
(PC2), inactive at the moment. The pulse is reflected from the mirror M2.
2. When PC1 (and PC2) is still switched off, the beam passes twice λ/4 plate and the pulse
is removed out of the amplifier by reflection from the crystal surface at the point A.
3. When PC1 is switched on (as a λ/4) before arriving the seed pulse, the total effect of the
λ/4 plate and PC1 is λ/4 + λ/4 + λ/4 + λ/4 = λ, which means that the polarization is
unaffected and the pulse is trapped inside, not reflected from the surface of the active
medium.
4. Each pass of the pulse through the active medium results in the pulse amplification.
When the pulse has been amplified to the desired level (~106 times), the voltage λ/4 is
applied to the Pockels cell PC2. The pulse going to the mirror M2 and returning through
191
PC2 (λ/4 + λ/4 = λ/2) changes its polarization and is removed from the amplifier at the
P point by reflection from the thin layer polarizer.
6.5. Chirped Pulse Amplification (CPA).
The configuration of the typical chirped pulse amplifier is presented in fig. 6.6. One
can see that it consists of the regenerative amplifier described in chapter 6.4 and the stretcher
and the compressor. The stretcher placed before the regenerative amplifier expands the pulse
duration by many orders of magnitude (from femtosecond to hundreds picosecond) and
thereby reduces peak power.
Employing the stretcher solves the problem of high intensities in the amplifiers for
ultrashort pulses that are above the damage threshold of the active medium of the amplifier.
For example, Ti:sapphire crystal has a high saturation fluence and a large gain-bandwidth
needed to generate relatively high energies for sub-picosecond pulses. Self-focusing in the
crystal described in chapter 3 is a desirable effect when employed in modelocking, but it has
also its dark sides. Beam focusing due to the Kerr lens focusing can lead to catastrophic beam
collapse that may destroy the crystal, which makes it necessary to limit the intensity present in
amplifiers to reasonable magnitude less than 10 GW/cm2. This obstacle can be removed by
the technique called chirped pulse amplification (CPA).
We have discussed in chapter 5 the factors that lead to stretching the temporal pulse
duration. The group velocity dispersion (GVD) is the most important factor affecting the
temporal pulse broadening. Due to the GVD each frequency component that comprises the
spectrum of the pulse travels through the medium with different group velocity. For so called
positive GVD materials, the red components travel faster than the blue components. As we
have shown a pair of prisms (fig. 5.28) can be used to compensate for the excess GVD that
results in the pulse compression. The same can be achieved with the diffraction gratings 6.20.
192
grating 1
re d e
u
bl
input
pulse
mirror
M
grating 2
output
pulse
(compressed)
Fig. 6.10 Illustration of the principle of the pulse compression
The pair of parallel gratings shows a simplified pulse compressor, which demonstrates
the concept of compression (fig. 6.10). One can see that the optical path through the grating
pair is longer for the longer wavelengths than for the shorter ones. The mirror M reflects the
beam back into the grating pair recovering the spatial distribution, but still increasing the
difference in the optical paths for the red and the blue components. Thus, the pair of the
parallel gratings provides the negative group-velocity. If the input pulse that is positively
chirped travels through the gratings, the output pulse becomes shorter due to partial
cancellation of the positive GVD effect by the negative GVD of the gratings in the
configuration presented in Fig. 6.10. Pulse stretching is essentially the reverse of the pulse
compression. The gratings can be configured in such a way so the bluer components have to
travel longer path through the stretcher than the redder components. The result is that the
stretcher generates a positive GVD effect that results in that the redder components travel even
faster than for the input pulse and exit the stretcher with the temporal pulse significantly
longer.
One of many possible configurations of the pair of gratings that generates the positive
GVD is presented in fig. 6.11. One can see that the main difference between the compressor
(fig. 6.10) and the stretcher (fig. 6.11) is that a telescope is added between the gratings to
invert the sign of the dispersion from the negative in fig. 6.10 to the positive in fig. 6.11.
193
1
mirr
or M
red
bl u
output
pulse
grati
ng
grating
input
pulse
e
lens
reverse GVD
2
Fig. 6.11 Illustration of the principle of pulse stretching
In practical applications some other configurations are used. For example, instead of using a
pair of gratings, a single grating combined with the curved mirror can be used as presented in
fig. 6.12. One can see that this multi-pass configuration reduces the complexity and ensures
four-pass the grating that is necessary for the spatial reconstruction of the stretched beam.
a
gr
3
1
tin
2
g
input
pulse
4
output
stretched
pulse
curved
mirror
Fig. 6.12 The pulse stretched with a single grating – curved mirror combination
The recent decade have witnessed the dramatic improvement in amplification of
ultrashort pulses. Amplification of the femtosecond pulses using the chirped pulse
amplification (CPA) technique 6.21 is now commercially available. Amplification of ultrashort
pulses to mJ, corresponding to multiterawatt peak powers becomes a routine task. CPA
technique is used both in the multipass configuration
6.25
6.22
and regenerative amplification
6.23-
. Quite often these configurations are combined together to reach terawatt powers. Usually
the regenerative amplifier is used as a first stage of amplification followed by multi-pass
configuration used for the output power-boost stage.
194
Generally, the multipass amplifiers are used for very short pulses (<50 fs). They need
much lower amount of optical material for amplification that reduce nonlinear distortion of
the pulse that usually comes at the expense of rather complicated alignment procedure and
higher level of ASE (amplified spontaneous emission). The regenerative amplifiers are
usually used for pulses longer then 50 fs. They have many advantages including much smaller
ASE and a simple operation and maintenance. No realignment is necessary to adjust the
number of passages through the amplifier since the beam follows the some path in the cavity.
So far we discussed the conventional configurations based on bulk-optic-solid-state
amplifiers. The development of optical communication created a growing market for higherpower optical amplifiers based entirely on the fiber technology. The development in doublecladding fibers technology and multi-mode diode pumping has increased cw fiber lasers
output to the level comparable with the solid-state lasers. However, achieving a few watts of
average power from fiber-based ultrashort laser systems, routinely produced by the solid-state
femtosecond systems, is not so straightforward. Limitations come from large effective
nonlinearities (SPM and GVD) described in chapter 5 that can destroy short femtosecond
pulses produced in the fiber core. The obstacles related to pulse amplification can be reduced
by using fibers with larger core. The typical core of a single-mode fibers are less than 10 µm,
larger cores operate usually in multimode regime. However, it has been found that it is
possible to obtain a single mode operation for a careful large-core fiber design 6.26. Increasing
an active area of the core enables an efficient pumping at shorter length of the fiber reducing
nonlinearities. Moreover, fibers exhibiting a positive GVD rather than negative GVD are used
for high-power amplifications. The soliton-like behavior (described in chapter 5) obtained
when the SPM and negative GVD offset each other at the properly chosen fiber lengths does
not apply at high powers. The typical optical amplifier is presented in fig. 6.13 6.27.
195
pump II
Yb-fiber pulse
source
pump I
Yb-fiber amplifier
chirp-control fiber
(GVD>0)
compressor
Fig. 6.13 Fiber laser amplification system
The system consists of a fiber-based ultrashort laser (Yb-doped fiber laser) that emits a
seed pulse for further amplification. The seed pulse (1050 nm, 2 ps, 300 mW, 50 MHz) passes
through a piece of a single-mode fiber exhibiting a positive GVD to stretch the seed pulse.
Then, the stretched linearly positively chirped seed pulse is amplified in a fiber amplifier. The
fiber amplifier (Yb-doped fiber, 4.3 m length, 25 µm-diameter core) is pumped from both
ends with two fiber-bundle-coupled laser-diode operating at 976 nm and 14 W. The seed
pulse is amplified to 13 W and the width increases from 2 ps to 5 ps. Then, the pulse is
recompressed down to 100 fs passing through a conventional diffraction grating compressor
(negative GVD designed) achieving 5 W output at 1050 nm.
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