APPLAuSE - Advanced Program in Plasma Science and Technology AT3. Introduction to lasers and laser-plasma interactions 1. Introduction to lasers (cont.) Gonçalo Figueira — goncalo.fi[email protected] Complexo Interdisciplinar, ext. 3375 Tel. 218 419 375
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APPLAuSE - Advanced Program in Plasma Science and Technology!
AT3. Introduction to lasers and laser-plasma interactions!
1. Introduction to lasers (cont.)!
Gonçalo Figueira — [email protected]!Complexo Interdisciplinar, ext. 3375!Tel. 218 419 375 !!
Today!1. Laser beam propagation and Gaussian
beams!2. Gaussian beam transformation!3. Spherical resonators!4. Pulsed lasers!
ESO's Compact Laser Guide Star Unit Ottobeuren, Germany www.eso.org
Many types of lasers emit beams with a Gaussian shape!
For all z, the amplitude (intensity) distribution along x and y is a Gaussian curve.!
software plot! real HeNe laser!
∝ exp − x2 + y 2
W 2(z )⎛⎝⎜
⎞⎠⎟
The Gaussian beam!
A(r) ≈ A1q(z )
exp −ik ρ2
2q(z )⎡
⎣⎢
⎤
⎦⎥
ρ2 = x 2 + y 2 q(z ) = z + iz0Gaussian beam: complex envelope!
q(z) !q-parameter of the beam!z0 !Rayleigh range!
!For convenience (we will see why soon) we can write q(z) as: !
1q(z )
= 1z + iz0
= 1R(z )
− i λπW 2(z )
Now let’s put everything together!The complex amplitude of the
Gaussian beam: U(r) = A(r) e-ikz!
€
U(r) = A0W0
W (z )exp −
ρ2
W 2(z )$
% &
'
( )
×exp −i kz − ζ(z )( )[ ]
×exp −ik ρ2
2R(z )$
% &
'
( )
€
A0 = A1 / iz0
W (z ) =W0 1+zz0
"
# $
%
& '
2
R(z ) = z 1+z0z
" # $
% & ' 2(
) * *
+
, - -
ζ(z ) = tan−1 zz0
W0 =λz0π
The beam is fully defined by A0 and z0 (and λ)!
amplitude!(real)!
phase!(imaginary)!
Gaussian beam: amplitude and phase!
€
U(r) = A0W0
W (z )exp −
ρ2
W 2(z )$
% &
'
( )
×exp −i kz − ζ(z )( )[ ]
×exp −ik ρ2
2R(z )$
% &
'
( )
Amplitude factor!describes beam spread!
Longitudinal phase factor!describes phase delay relative to a plane wave or spherical wave!
Radial phase factor!phase shift due to measuring a spherical surface along a plane!
Gaussian beam: amplitude and phase!
€
U(r) = A0W0
W (z )exp −
ρ2
W 2(z )$
% &
'
( )
×exp −i kz − ζ(z )( )[ ]
×exp −ik ρ2
2R(z )$
% &
'
( )
Amplitude factor!describes beam spread!
Longitudinal phase factor!describes phase delay relative to a plane wave or spherical wave!
Radial phase factor!phase shift due to measuring a spherical surface along a plane!
Properties of the Gaussian beam!The intensity is given by:!
€
I(ρ,z ) = U(r) 2 = I0W0
W (z )#
$ %
&
' (
2
exp −2ρ2
W 2(z )#
$ %
&
' ( I0 = Ao
2( )• For any z the intensity is a Gaussian function of ρ!
• The Gaussian function has its peak on the z axis (ρ=0), and decreases monotonically as ρ increases.!
• W(z) is the beam width of the Gaussian distribution; it increases with the axial distance z.!
• On the beam axis the intensity is!
€
I(0,z ) = I0W0
W (z )"
# $
%
& '
2
=I0
1+ z / z0( )2
Intensity vs. distance z!
€
I(0,z ) ≈ I0 z0 / z( )2, z >> z0
NB this one is not a gaussian!
z = 0 I = I0
z = z0 I = I0/2
z = 2 z0 I = I0/5
x
y
Gaussian beam: power!Using the definition for optical power,!
€
P(z ) = I(ρ,z )dAA∫ = I(ρ,z )2πρdρ
0
∞
∫= 1
2 I0 πW02( )
• Total power = (half the peak intensity) × (the beam area) !• Independent of z (i.e. energy is conserved)!
A circle of radius W(z) contains ~86% of the total power!
power in circle of radius ρ0 total power!
= 1P
I(ρ,z )2πρd ρ0
ρ0∫
= 1− exp − 2ρ02
W 2 z( )⎡
⎣⎢
⎤
⎦⎥
A circle of radius ρ = 1.5 W(z) contains 99% of the total power!
W(z) is the beam radius,with a minimum W0 at z = 0!
ρ = W(z) = beam radius or beam width!
€
W (z ) =W0 1+zz0
"
# $
%
& '
2
• Minimum value (beam waist) happens for z = 0: W(0) = W0!• Waist diameter 2W0 is called spot size!• Beam width has the value √2W0 at z = z0!• Width increases linearly for z >> z0!
Gaussian beam: amplitude and phase!
€
U(r) = A0W0
W (z )exp −
ρ2
W 2(z )$
% &
'
( )
×exp −i kz − ζ(z )( )[ ]
×exp −ik ρ2
2R(z )$
% &
'
( )
Amplitude factor!describes beam spread!
Longitudinal phase factor!describes phase delay relative to a plane wave or spherical wave!
Radial phase factor!phase shift due to measuring a spherical surface along a plane!
The phase of a Gaussian beam has three components!
€
ϕ(ρ,z ) = kz − ζ(z )[ ] + k ρ2
2R(z )
kz !phase of a plane wave propagating along z!!
!phase delay specific of the Gaussian beam, that makes!it different from either a plane or a spherical wave.!This is called the Gouy effect.!
! term responsible for wavefront bending i.e. shift from a plane to a spherical wavefront at off-axis points.
!
The phase components ζ(z) and kρ2/2R(z) vary slowly with z !
€
ϕ(ρ,z ) = kz − ζ(z )[ ] + k ρ2
2R(z )
1/R(z)
ζ(z)
Gaussian beam: wavefronts!
€
ϕ(ρ,z ) = k z +ρ2
2R(z )$
% &
'
( ) − ζ(z ) = 2πq
Since ζ(z) and R(z) vary slowly, they may be considered constant for low values of ρ: ζ(z) ≈ ζ and R(z) ≈ R, leading to!
z + ρ2
2R≈ qλ + ζλ / 2π
This represents a paraboloidal surface of radius of curvature R = R(z)!
R(z ) = z 1+ z0z
⎛⎝⎜
⎞⎠⎟2⎡
⎣⎢⎢
⎤
⎦⎥⎥
R(z) is the radius of curvature.It has a minimum ±2z0 at z = ±z0.!
Note that the sign convention for wavefronts and for optical surfaces is the opposite!
Viewing a Gaussian beam propagation!
(YouTube, Propagation of a Gaussian beam, computed with a FDTD code)
On the other hand:!!!so we have!
A Gaussian beam may be described by its complex q-parameter!
If we know q(z):!!then!
€
q(z ) = z + iz0
€
1q(z )
=1
R(z )− i λ
πW 2(z )€
Re q(z )[ ] = z = distance to beam waist
Im q(z )[ ] = z0 = Rayleigh length
R(z ) = Re 1 q(z )[ ]−1 = radius of curvature
W (z ) = λπ
Im − 1q(z )
⎡⎣⎢
⎤⎦⎥
−1
= beam radius
Transmission through an arbitrary optical system!
Remember the ABCD matrices from ray optics? Consider an arbitrary paraxial optical system characterized by an [ABCD] matrix:!
€
q2 =Aq1 +BCq1 +D
The ABCD law!
The same rules for cascading optical components apply.!
Spherical mirror resonators!
�
R1 < 0, R2 < 0Pay attention! In this case we have:
Conditions for ray confinement!In ray matrix theory we studied the conditions for a bounded solution in a periodic system.!
For a spherical mirror we have the confinement condition!
�
0 ≤ g1g2 ≤ 1, g1,2 = 1+ dR1,2
⎛
⎝ ⎜ ⎞
⎠ ⎟
�
0 ≤ g1g2 ≤ 1
g1g2< 0> 1
⎧ ⎨ ⎩
g1g2= 0= 1
⎧ ⎨ ⎩
stable resonator!
unstable resonator!
conditionally stable resonator!
The stability depends on the product of the g-parameters!
Diagram of resonator stability!
�
g1,2 = 1+ dR1,2
⎛
⎝ ⎜ ⎞
⎠ ⎟
R g ∞ 1 -d 0
-d/2 -1
The dotted red line marks the position of symmetric resonators for which g1 = g2
Symmetric resonators: confinement condition!
In this case R1=R2=R and g1=g2=g :!
�
0 ≤ g2 ≤ 1 ⇔ 0 ≤ d(−R)
≤ 2
A stable symmetric resonator must use mirrors with a radius of curvature greater than (length/2).!Example: symmetric confocal ( R = -d )!
Gaussian beams: quick reminder!
�
z0 = πW02
λ
W (z ) =W0 1+ zz0
⎛ ⎝ ⎜
⎞ ⎠ ⎟ 2
R(z ) = z 1+ z0z
⎛ ⎝ ⎜
⎞ ⎠ ⎟ 2⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
ζ(z ) = tan−1 zz0
W0 = λz0π
�
I (ρ,z ) = I0W0
W (z )exp − ρ2
W 2(z )⎡
⎣ ⎢
⎤
⎦ ⎥
×exp −i kz − ζ(z )( )[ ]×exp −ik ρ2
2R(z )⎡
⎣ ⎢
⎤
⎦ ⎥
The Gaussian beam is a mode of the spherical mirror resonator!
A Gausian beam will be reflected at a spherical mirror and retrace its way back exactly if:!
Rwavefront = Rmirror!
Parameters of the Gaussian beam that obeys the boundary conditions:!
Conditions for Gaussian beam confinement!
We have the following three conditions linking the positions z and radius R1, R2 of the concave mirrors, and the R(z) of the beam:!
!!!!!
�
z2 = z1 + dR1 = z1 + z0
2 / z1(−R2 ) = z2 + z0
2 / z2
�
z1 = −d R2 +d( )R2 +R1+2d
z02 = −d R1+d( ) R2 +d( ) R1+R2 +d( )
R2 +R1+2d( )2
W0 = λz0 / π
Wi =W0 1+ ziz0
⎛⎝⎜
⎞⎠⎟
2
, i = 1,2
R2<0, R(z)>0 R1<0, R(z)<0
This gives us the position of the beam center and Rayleigh length:
Why pulsed lasers?!Power = Energy / time!
!Concentrating the optical energy in a (short)
pulse increases the power of the laser.!1 µs !microsecond !10-6 s !0.3 km !!1 ns !nanosecond !10-9 s !0.3 m!1 ps !picosecond !10-12 s !0.3 mm!1 fs !femtosecond !10-15 s !0.3 µm!1 as !attosecond !10-18 s !0.3 nm!
Ultrashort laser pulses are the shortest events produced by mankind!
Progress in short-pulse generation!
© A
lexa
ndra
A. A
mor
im, F
CU
P!
Methods of pulsing lasers!External modulation!
CW laser + ext. switch (on/off)!!This is inefficient and the output
power is only the CW power!!!!
Internal modulation !Energy is stored (off) and
released (on)!Efficient and allows very high
peak power!
Gain switching!The gain is controlled by turning the pump on and off!Examples:!• flashlamp-pumped Ruby laser!• pulsed semiconductor lasers (pump = electric current)!
gain switching
Q-switching!The output is modulated by switching on/off the resonator losses (Q
parameter): Q-switching = loss switching!• energy is stored in the form of inverted population!• when losses are reduced, “giant” laser pulses result!!
Q-switching
Example: Spectra Physics Quanta-Ray Nd:YAG laser!
Q-switching is achieved by a combination of a polarizer + fast polarization rotator (Pockels cell)!
!Specifications:!• E (1064 nm) up to 2.5 J!• Pulse duration ~ 6 – 12 ns!• Peak power up to ~ 0.4 GW!• Rep rate: 10 – 100 Hz!
Cavity dumping!The output is modulated by switching on/off the output mirror
transmittance!• photons are stored inside the resonator!• output results from e.g. removing the mirror (T = 100%)!!
Cavity dumping
Mode locking!It’s a dynamic steady-state process!• pulses result from coupling the resonator modes together and
locking their phases !• the output consists in a periodic pulse train of period 2d/c!!
Mode locking
How mode-locking works!
Random phases
Light bulb
Intensity vs. time
Short pulse
Locked phases
Time
Time
Intensity vs. time
Modes of a laser resonator!
€
U(r) = 0
€
U(r) = A sin(kz ), kd = qπ (q = 1,2,…)
kq = q πd
Uq (r) = Aq sin(kqz )
The modes are standing waves and q = 1,2… is the mode number!
€
ν = c λ =kc2π
kq = q πd
⇒ ν ≡ νq = q c2d
resonance frequencies
€
νF =c2d
=1TF
frequency spacing and period
Light inside a resonator is the sum of all possible modes!
Let’s consider each mode as a plane wave propagating in the z-direction with velocity c = c0/n:!
€
U(z,t ) = Aq exp i2πνq t − zc
% & '
( ) *
+
, -
.
/ 0
q∑
νq = ν0 +qνF , q = 0,±1,±2,…
complex envelope (amplitude and phase)!
modes
central frequency
The spacing of the modes νq is determined by the resonator dimensions.!The magnitude |Aq| is set by the gain curve of the laser medium and resonator losses.!In general, the phases arg{Aq} of each mode are independent.!
Adding all the modes!
€
U(z,t ) = A t − zc
# $ %
& ' ( exp i2πν0 t −
zc
# $ %
& ' (
+
, -
.
/ 0
A(t ) = Aq expq∑ iq 2πt
TF
#
$ %
&
' (
TF = 1/ νF = 2d /c
Wave travelling along z-axis at speed c with frequency ν0. Periodic in z with period cTF = 2d.
Sum of individual plane waves of complex amplitudes Aq and frequencies qνF. A(t) has period TF.
If the complex amplitudes Aq are random, U(z,t) is just periodic noise. But if they are adjusted, U(z,t) can become a periodic train of powerful pulses: mode locking means arranging the modes so that this happens.
We can “lock” the modes by controlling their phase!
Let’s assume for instance that all Aq are equal: Aq = A.For M = 2N+1 modes we have!
€
A(t ) = A exp iq 2πtTF
#
$ %
&
' (
q=−N
N
∑ = A xqq=−N
N
∑
€
A(t ) = Asin Mπt /TF( )sin πt /TF( )
The sum of the series can then be calculated:!
€
I(t,z )∝ A(t ) 2
M = total number of modes
Larger M allows shorter and more powerful pulses!
M is proportional to the linewidth Δν,!so the pulse duration τpulse is
proportional to 1/Δν:!
€
M ≈ Δν / νFτpulse =TF /M ≈ 1/Δν
The peak intensity is proportional to (M × average intensity):!!
€
I = A 2∑ = M A 2
Ip = limt→TF
A 2 sin Mπt /TF( )sin πt /TF( )[ ]
= M 2 A 2= MI
Mode locked pulse train: summary!
When the resonator modes are locked in phase, the output is in the form of a train of short, powerful pulses.!
€
TF =2dc
€
cTF = 2d
€
I
€
Ip = MI €
cτp =2dM
€
τp =TFM
=1Δν
Temporal period!
Spatial period!
Mean intensity! Peak intensity!
Pulse length!
Pulse duration!
How to achieve mode locking?!
Active mode locking!An amplitude (AM) or phase (FM) modulator is placed inside the cavity, e.g.!!• acousto-optic switch!• electro-optic switch!!
Passive mode locking!A component behaving as a saturable absorber is used: losses are high for low intensity light and low for high intensity.!!• SESAM (semiconductor
saturable absorber mirror)!• Kerr-lens mode locking!
By placing a component inside the laser cavity that discriminates “good” (high intensity) from “bad” (high intensity) mode superposition.
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