Upload
asghar-farhadi
View
14
Download
0
Embed Size (px)
DESCRIPTION
Laser
Citation preview
201
Rate Equation forRate Equation for22--Level SystemLevel System
2St.em. Sp.em. Stim.abs.
221 2 21 2
12 1
'
'
dNR R R
dt
dN IB g v N A N
dt c
IB g v N
c
2
Rate equation for a 2 - levelsystem is the change inoccupance of N per time :
202
Rate Equation forRate Equation forEnergy Level Energy Level ““11””
1St.em. Sp.em. St.abs.
121 2 21 2
12 1
'
'
dNR R R
dt
dN IB g v N A N
dt c
IB g v N
c
203
Capture CrossCapture Cross--SectionSection
12 21
21
Capturecross-section
area
Equation 26-9
'' area
B B
hvB g v
c
Define Capture Cross - Section :
204
Rewriting Rate Equation Rewriting Rate Equation for Twofor Two--Level System:Level System:
221 2 2 1
121 2 2 1
1 2
'
'
T
dN IA N N N
dt hv
dN IA N N N
dt hv
N N N
Total # of atoms is :
205
TwoTwo--Level Laser canLevel Laser can’’t t occur because:occur because:
Inversion can’t happen:
Ninv = N2 – N1
= Negative value
Stim. Emiss. + Spon. Emiss.> Stim. Absorp.
Since Capture Cross Section (σ)is the same!
206
Two (2) Common Laser Two (2) Common Laser SystemsSystems
3 Level
E3
E2
K21 - Lasing
N2
E1
Pump
K32 spon
N1
p
p
I
hv
Nt =N1+ N2+ N3
3 level
4 level
207
22ndnd Common Laser Common Laser SystemsSystems
4 Level
3
2
Laser
N2
0
N11
NT = N1 + N2 + N3 + N0
208
44--Level Laser SystemLevel Laser System
3
2
Rate → Laser λ
N2
0
N11Ip
N3
K32
K21
K10
N0
32 21
32 3
3
p
is very fast so e don't reside in .
0
In fact, the assumption is that e in 3 instantaneouslyfalls to level 2.
Figure 26-5, p. 557 (Pedrotti).
Photon from pump must have a capturecros
K K
K N
N
hv
s section.
1
Lifetime
Pumping Rate
Stimulatedemission
p
K
I
R I
K20
K31
K30
209
Rate Equations for Pumping a Four Rate Equations for Pumping a Four Energy Level System: Change in Energy Level System: Change in
Occupy per TimeOccupy per Time
33 3 3 0
232 3 2 2 2 1'
is
p p
p
IdNK N N N
dt hv
dN IK N K N N N
dt hv
I
View 4 - level structure and write down rate ofchange for each level
Gain from ground
Population Change, Level 3, #/sec
Stimulated emission
131 3 21 2 10 1 2 1
030 3 20 2 10 1 3 0
T 0 1 2 3
3 30 31 32
2 21 20
laser cavity Irradiance
d
dt '
p p
p
N IK N K N K N N N
hv
IdNK N K N K N N N
dt hv
N N N N N
K K K K
K K K
All the atoms in Four Levels :
210
Rate Equations for Pumping a Rate Equations for Pumping a Four Energy Level System: Four Energy Level System: Change in Occupy per TimeChange in Occupy per Time
/3 2 1
3 2 10
2p
2
1 1 1 1; ; ; point
Capture cross section M for pump photons
watt Pump irradiance,
M
Energy of photons used in pump (joules)
Capture cr
t
p
p
eK K K e
I
hv
Recall K are reciprocal lifetimes :
oss section for Lasing photons @ 'v
211
44--Level Laser OperationsLevel Laser Operations
2p 0
0 3 0 0 1 2 3
30
3 3 0 33
232
Pump power flux (watt/M ) I does not empty the ground level (N )
; so
0 in steady state population inversion;
1
0
T
RT
p p p pT
p
N N N N N N N N
dNI I
dtI I
K N N N Nhv K hv
dNK
d
Assumption :
3 2 2 2 1
322 2 2 1
3
2
31121 2 10 1 2 1
3
1
2 1
'
'
relate to pump power input
0'
Solve for for the gain or inversion
p pT
p
p
p pT
p
p
IN K N N N
hv
IK IN K N N N
K hv hv
R
IKdN IN K N K N N N
dt K hv hv
R
N N
21 22 1
10 102 1
202
10
population
1
1'
"Population Inversion" Function of optical Power
p p
K KR R
K KN N
K IK
K hv
212
Absorption/Loss Coefficient, Absorption/Loss Coefficient, αα
2
Use this inversion to produce gain in thethe optical irradiance as it propagates. The
incremental change in irradiance (w/M ) is( ) in time going through a volume
w
dII
dz
I V
Change in irradiance :
ith cross section and length
'
# of photons generated
I '
z
A z
hv nI
A
n
hv n
V t
Dividing by Δz :
I + ΔII
ΔA
213
Change in Irradiance Change in Irradiance through a Volumethrough a Volume
2 1
22 1
'
Each transition produces '
Volume of the gain media
Rate of change eq. of photons
is equal to population rate ( )
( ) slide 205'
I n hv
z t V
hv
V
N N
n IN N V
t h
214
Absorption CoefficientAbsorption Coefficient
2 1
2 1
2 1 2 1
differential
'
Rate of stimulated emission
Substitute into
'
'
But recall 210; definition
n IN N V
t hv
I
z
I hv IN N V
z V hv
dII N N N N I
dz
dI
dz
2 1
Their absorption is negative
Where is gain coefficient
I
N N
dII
dZ
215
Typically:Typically:
0 0
0
0
ln
Beer's Law
LI L
I
L
LL
dII
dz
dIdz
I
IL
I
I I e
216
Gain CoefficientGain Coefficient
2 1
0
Gain coefficient
LL
N N
I I e
How much pumppower is needed?
217
Laser Power Output Laser Power Output ConditionsConditions
• Set up 2-mirror laser cavity
• Derive output irradiance
– eq. 26-47(Pedrotti, derived inSlides 220-224)
• Stability conditions
• Homogeneous vs. inhomogeneous broadening
218
Optical TransitionsOptical Transitions
E2 + ΔE2
E1 + ΔE1
γ
Gain Curveγ = σ(N2 – N1)
Max gain for hv = E2 – E1
v
Lasing Energy Levels
Band of energy levels
(E2 – E1)
(E2 + ΔE2) –(E1 – ΔE1)
Corresponds to
(E2 – ΔE2) –(E1 + ΔE1)
219
Modes of LaserModes of Laser
γ
Gain Curve
γ = σ(N2 – N1)
Max gain for hv = E2 – E1
v
3 Modes must be above gain threshold
2
C
L 2
C
L
220
Laser CavityLaser Cavity
0
Mirror (100% refl.)output mirror (<100%)
cavity lengthgain media length
Resonants
sin
Electric field must be phase exactlyafter round trip, within cavity
22 2 2 ;
integer
2L
2
L
LL
E E kz t
k L L m
m
m
Lv
c
A.
B.
2
resonants occur @c/2 apart and are called modes
m
cv m
L
L
221
Threshold ConditionThreshold ConditionCW CW –– Continuous WaveContinuous Wave
2 2 2 1R l
Scattering, etc.
0
00 0
0 2 1
2
2 10
;
; gain coefficient
Round trip intensity from initial location
;
Small signal output (1% coupler); steady state conditi
g
Lg
Lg
LRT
dI dII dz
dz I
I I e N N
IR R e Milonni & Eberly (Wiley, 1988)
I
00
2
2 10
1 2 1 22 1
1 2
onwarrants that the intensity remains constant inside cavity
; 1
1 ; small signal gain coefficient
1ln 2 ln 1 ln
Trig expression, 1 Taylor Exp.
T g
RTRT
LRT
T g
II I
I
IR R e
I
L R R R RR R
R R x
T
1 2
1 2
Steady state - threshhold gain. coefficient - 1
1 other losses per length2
1 1Threshhold gain ln
2
Tg
T Tg
x
R RL
L R R
222
Ideal 4 Level LaserIdeal 4 Level Laser
31 202 1 p1
3 10
10 10
No simulated emission transitions from 3 to 1 level;k
0 in Equation (207) of force R to z; 0
Atoms in energy level 1 fall to zero (ground state) with zerolifetime, 1/
Optic
kN N
k k
k
2 1
02 2
2
2 320 2 2 2
2 3
2
0
al Gain - Equation 26-37 - from slide 207,
gain 1 ' 1
Where we grouped terms:
'Saturation irradiance
Setting threshold gain
P
s
p p pp T
p
s
N N
R kII hv k
I
R IkR N
k k hv
hvI
0
1 2 Sat.
0
Sat. 1 2
0Sat.
1 2
equal to increase population gainand equal above,
1 1ln
2 1
21
ln 1
21 Irradiance/intensity
ln 1 inside cavity
T
Tg
g
g
L R R I I
LI
I R R
LI I
R R
223
Laser OutputLaser Output
As we increase γ0 (gain coeff.), it reaches γthreshhold and can’t get above γT
out 2 2
; ; 2
Positive z direction field2 gives output beam
Substitute from previous slide
I I I I I I I
II T I T
I
20pumps p
KI
320 2
3
pumps (Slide 217)p Tp T p
p
NkI N I
k hv
←Transmissionof Mirror 2
224
Laser OutputLaser Output
1 2
0 Sat.out 2
1 2
1 2 1 21 2
1 2
1 2 2
1 2 2
1ln 1 Taylor expansion
21
1 2ln
1ln ln1 ln ln
1
1 Loss due tomirror's transmission
+
is 100%, so 1 but there
g
x R Rx x
L II T
R R
R R R RR R
R R
R R T
R R T
0 satout 2
2
are other lossessuch as scattering and spontaneous emission
21
2gL I
I TT
225
Output Power DerivationOutput Power Derivation
Pedrotti3 26-19. Derive Eq. (26-47) by a procedure similar to that leading to Eq. (26-43).
The linear cavity case is complicated by the fact that the field encounters the gain medium twice in each round-trip with the losses encountered at the mirrors interspersed between passes through the gain medium.
It may be useful to research and then summarize the solution to this problem.
Gain Medium
IoutI2+I1+
I1- I2-
R1 R2, T2L
Right going field: Left going field: dI dI
I Idz dz
226
Output Power Derivation, cont.Output Power Derivation, cont.
0
Saturation is due to both the right and left going fieldsin the cavity, so we can rewrite Equation (26-40) as:
1 1(1)
1
1 1This relation leads to: 0
Where the abo
S
dI dII II dz I dz
I
dI dI
I dz I dz
2 2
ve expression is used below:
1 1 1ln
ln 0
Therefore: ln
Now find the output irradiance shown in Eq. 26-47.
(2)
(3)
(4)
Solve (
C
out
dI dIdI I
dz I dz I dz I I
dI I
dz
I I C I I e C
I I I
I I C
I T I
3) for and substitute into (2): C
I I II
227
Output Power Derivation, cont.Output Power Derivation, cont.
2
1
+
0
0
0
22 1 0
1 2 1
1 1
Take (1) for the I equation, separate variables and integrate.
1
1
11
1 1 1ln (5)
Now,
S
I L
SI
S S
dII II dz
I
CI
IdI dz
I I
I CI I L
I I I I I
I I
2 2
1 2
1 1 1
2 2 2
1 1 2 2
11 2 2 2
1
1 2 1 2 2 2
and are the reflectance of the two mirrors, so that
Using the above 3 relationships:
Therefore: and
I I C
R R
R I I
R I I
I I I I
II I I R
R
I I R R C I R
228
Example Problem, cont.Example Problem, cont.
22 2 1 2
2 1 2
2 20
2 2 1 2
2
2 2 21 2 2 0
11 2
2 21 2 2 0
1 1 2
Substituting these into (5):
1ln
1 1
Solve for I :
1ln 1
1 11 ln
2
S
S
S S
S
II I R R
II R R
I RL
I I I R R
I I RR R R L
I I RR R
I RR R R L
I R R R
2 21 2 1 2 0
1 1 2
2 21 2 0
1 1 2
01 22
21 2
1
1 11 1 2 ln
2
1 11 1 2 ln
2
12 ln
21 1
S
S
Sout
I RR R R R L
I R R R
I RR R L
I R R R
LR RT I
IR
R RR
-- Derivation courtesy
of Kyung Lyong Jang
229
2 Mirror Laser2 Mirror Laser2
1
1
0 Sat.out 2
2
Mirror 2 is output coupler - T ; from slide 219
Therefore T is a loss if reflectance is not 100%
Also, other losses in cavity
All losses = T scattering absorp.
21
2
iff losse
gL II T
T
2
2 2 2
Sat.out 0 2
0
out Sat.
Sat.
s are much less than T
,
22
Output varies (depends) linearly on
(Small signal gain coefficient).
Different materials have different ( ) saturation lifetimes.
g
T T T
II L T
I I
I
21
Sat.0 0 21
Lifetime
1
2 2
Higher ultraviolet, x-ray, are more difficult to build.
k hvI
v
2
1
230
0
0 2 1
Small signal gain coefficient depends on
population inversion,
#Pump rate
Sec.
i.e., 3 Level Laser
N
N N
N
Pumprate
Fast decayN2
N1
gain
γ
γT
Threshhold
Gain saturationround trip intensity in
cavity is constant
Pump rate
Pout
2
out
Sat.out 0
out
opt 0
What is the optimum output coupler ( )?
Solve for given loss:
22
0
2
g
g
T
I
ITI L T
T S
I
T
T L
Iout
ToptT2
Outputcoupler
x
231
Laser Cavity Stability ConditionsLaser Cavity Stability Conditions
1 2
1 11
22
1 2
1 2
1 2
1 2
0 1
1 ; radius of curvature of mirror
1
Confocal cavity
1 1 1 1 0
Concentric Cavity
2
1 2 1 2 1
These are li
g g
Lg r
r
Lg
r
L r r
g g
L r
r r r
g g
Two extremes to get Lasing :
A.
B.
mits to cavity length!
232
Laser Line Width orLaser Line Width orLineshapeLineshape g(g(vv’’))
• Not perfectly monochromatic
Two general mechanisms that cause freq. broadening:
– Homogeneous – Physical phenomena in gain media that affect all atoms in the same manner that broaden the frequency response of each atom
– Inhomogeneous – Physical mechanism that affects different groups of atoms in different manners, i.e., He or Ne atoms
Full width at half maxHv ½
233
•Lifetime Broadening– τ2 is not exact value, but has some spread
– Causes:A. Spontaneous emission
B. Inelastic collision of atoms, which changes its energy (Pressure, temperature of media)
Homogeneous BroadeningHomogeneous Broadening
•Pressure Broadening– Elastic collisions of atoms cause delays in emission
– Typically the dominant cause of homogeneous bonding (vcol)
1 2 Time
τL + ε τL + ε2
2
2
0
col2 1
2 2 1 1 col
Lorentzian line shape
22
1 1 1 2
2
Lifetime , Lifetime , Rate of collisions
H
H
H
g v
vg v
vv v
v v
E E v
Gain Bandwidth :
234
Recall Gain Cavity:
0Electronic field sin .Round trip has to be in phase (exactly).
22 2 4 2 ( ) ( is integer)
4 2 ; 0, 1, 2;2
Therefore, cavity will have/support any
frequency that ha
E E kz t
vkz k L L L kz m m
cv c
L m m v mc L
v
8
8
s values or modes2
Modes spacing in frequency is 1; apart2
3 10i.e., 1 meter, 1.5 10 Hz
2 1
Homogeneous broadening gain media only allowes lasingat one of these frequencies - single cavity mod
cm
L
cm
L
L v
e!
235
Homogeneous Gain MediaHomogeneous Gain Media
Only one mode in cavitywith homogeneous broadening
236
Inhomogeneous BroadeningInhomogeneous Broadening
• Doppler effects (gas laser) causing larger linewidth
–Velocity of Ne atoms are moving with a distribution of velocities which produce a doppler shift in frequency, v (train whistle).
–Velocity of Ne atoms are moving in 4π steradians so only zcomponent is causing frequency shift
237
InhomogenousInhomogenous BroadeningBroadening
20
14 ln 22
2
1
2
02
26
4 ln 2
Full width at half max
8ln 2
Mass of atom (atomic mass)
6.64 10 kg
D
v v
v
D
D
BD
g v ev
v
k Tv v
Mc
M
Gaussian Lineshape :
238
Inhomogeneous BroadeningInhomogeneous Broadening
Δvinhomo.> Δvhomo.
Willis Lamb - @OSC Professor
239
High Intensity is Possible High Intensity is Possible by Pulsed Operationby Pulsed Operation
– Super saturate the cavity; then, switch it on for a short time• Q switch• Mode lock
– Cause population inversion but don’t let it lase
– Average power = continuous wave (CW) operation
240
Laser Types & WavelengthLaser Types & Wavelength• Gas
– HeNe, CO2
• Solid State– Nd:YAG– Ruby
• Chemical– HF & DF
• Free-Electron– X-ray
• Range– 1nm < λ < 1mm
• Continuous wave – CW Power– 1 mw to 5 megawatts(?)
• Pulsed Power– 1015 joules
• Pulsed Length, shortest– 5 fsec → f = 10-15
• Cavity Lengths – L– Few μm < L < km
241
LED – Light Emitting Diodes
• Emitted photon goes in 4π steradians
--Courtesy of Tomasz Tkaczy
242
Surface Emitting LED
--Courtesy of Tomasz Tkaczy
243
Laser Diodes –Electrical & Optical Properties
• Cavities, mode behavior
• Power – current plot
• Divergence
• Astigmatism
• Polarization
• Laser Diodes as geometrical light sources
--Courtesy of Tomasz Tkaczy
244
Laser Diode Electronic Properties
--Courtesy of Tomasz Tkaczy
245
Laser Diode Cavities
--Courtesy of Tomasz Tkaczy
246
Laser Diode Power-Current Typical Curve
--Courtesy of Tomasz Tkaczy
247
Laser Diode Mode Behavior
--Courtesy of Tomasz Tkaczy
248
Laser Diode Polarization
--Courtesy of Tomasz Tkaczy
249
Laser Diode Divergence
--Courtesy of Tomasz Tkaczy
250
Laser Diode Divergence
--Courtesy of Tomasz Tkaczy
251
Circularize by single-prism expansion
--Courtesy of Tomasz Tkaczy
252
Gaussian Beam ProfilingGaussian Beam Profilingand Propagationand Propagation
• Objectives:
– Review on Gaussian beam intensity profile
– Geometrical approach to Gaussian beam profiling
– Beam propagation
253
Classical Treatment of Classical Treatment of Gaussian BeamGaussian Beam
• Gaussian beam: Ideally, the irradiance distribution in any transverse plane is a circularly symmetric Gaussian function entered about the beam axis
• 64% between std. dev.• 86.5% energy between ±w
254
Classical Treatment of Classical Treatment of Gaussian Beam (Cont.)Gaussian Beam (Cont.)
W0 = Beam Waist; smallest radius in that space
Measure distances relative to beam waist@z = 0
255
Classical Treatment of Gaussian Classical Treatment of Gaussian Beam (Cont.)Beam (Cont.)
• The complex electric field amplitude of Gaussian beam (eq. 27-24 Pedrotti):
2 210 0
02( , ) exp exp tan ( / )
( ) ( ) 2 ( )
E WE z i kz k z z t
w z w z R z
Amplitude Phase
Plane wave
Spherical wave
Phase retardationθ = -π/2 at z = -∞θ = π/2 at z = ∞
2 2
2: Wave numberk
x y
2 22 0
0 2
2
0 0
Beam irradiance:
2( , ) ( , ) exp
( ) ( )
WI z E z I
w z w z
I E
256
Rayleigh RangeRayleigh Range
20
0
Given without proof:See equation 27-21
Wz
Figure 27-2 (Pedrotti)
257
Classical Treatment of Gaussian Classical Treatment of Gaussian Beam (Cont.)Beam (Cont.)
0
0
20
0 0 0
2
0 0 20
The axial distance within whichthe beam Intensity drops to ½ and beam radius lies within
a factor of 2 of waist radius
( ) 2
1( , ) exp
2
W
Wz w z W
I z IW
Rayleigh range (Z ) :
Beam diverge
20 0
00 0
*angular spread is fn(wavelength)
and waist is small -- recall Fourier
W*tan if z , we have ( )
Wz w z z
z W
nce angle :
258
Classical Treatment of Gaussian Classical Treatment of Gaussian Beam (Cont.)Beam (Cont.)
• Definition: Beam radius or beam width w(z): the radial distance of a circle that contains approximately 86% of the power (or irradiance drops down to 1/e2
13.5% of the peak irradiance)2 2
2
00 0 02
0 0
( ) 1 1Wz z
w z W W zz W
• Beam waist W0: where beam radius is minimum, i.e. z = 0 plane– The wavefronts are approximately planar near the beam waist
(See page 239)
x
y
W(z)
20 0
0 0 2 220
0 0
0,1
W zI z I I
z zW z z
2 20
0 2
2,
WI z I e
W z W z
259
Classical Treatment of Classical Treatment of Gaussian Beam (Cont.)Gaussian Beam (Cont.)
22 20 0
220
0 0 0
10
0
Radius of wavefrontat the Rayleigh waist
Phase
( ) 1 1
@ Rayleigh distance
2( ) 2
( ) tan ( ) 4
z WR z z z
z z
WR z z kW
zz z
z
Radius of Curvature :
Phase retardation :
2 20 0
2
at the Rayleigh waist
( , ) exp exp ( )( ) ( ) 2 ( )
E WE z i kz k z
w z w z R z
260
Classical Treatment of Gaussian Classical Treatment of Gaussian Beam (Cont.)Beam (Cont.)
2 20
0 2
2
0 00 2
0
2( , ) exp
( ) ( )
(0, )( ) 1 ( )
WI z I
w z w z
W II z I
w z z z
Intensity on z axis :
0 029 2 2
1
I I
zz
0
2
I
261
Classical Treatment of Classical Treatment of Gaussian Beam (Cont.)Gaussian Beam (Cont.)
20 0
0
2 20
0 2
Beam power: the total power, , carried by a beam is the
integral of the optical irradiance over a given transverse
plane at a distance z:
1( , )2
2
2( , ) exp
( ) ( )
I z d I W
WI z I
w z w z
0
2
2 2
0
0
0
20
2
2 2exp
( ) ( )
Relative power carried by an aperture with a radius r
1( , ) ( , )2
21 exp
( )
r
w z w z
p z I z d
r
w z
x
y
ρ0
262
Classical Treatment of Classical Treatment of Gaussian Beam (Cont.)Gaussian Beam (Cont.)
0 20
0 20
2
Encircled energy by the Beam width w(z) (radius)normalized contains approximately 86% of thetotal power:
21( , ) ( , )2 1 exp
( )
1, 1 86%
1.5 , 99%
p z I z dw z
p w z ze
p w z z
x
y
ρ0
w0=1
263
Summary of Gaussian Beam Eq.Summary of Gaussian Beam Eq.
2
2 2
20 0
2 20
0 2
20
0
1/ 2 22
0 020 0
2 2Irradiance: ( , ) exp
( ) ( )
1watts
2
2( , ) exp
( ) ( )
Rayleigh range
( ) 1 1
I zw z w z
I W
WI z I
w z w z
Wz
z zw z W W
W z
1 1 λ= +iq(z) R(z) πw(z)
1/ 2
0 0
2 220 0
0
0 0
( ) 2
( ) 1 1
tan ff
w z W
W zR z z z
z z
W
z W
264
Example ProblemExample Problem
• 27-23 The output from a single mode TEM00 Ar+ laser ( λ = 488 nm) has a far field divergence angle of 1 mradand output power of 5 watts.
0
3
0 3
20 0
20 2 2
0
0
tan
0.488(10 )
(10 )
15 watts
2
2 2 51.32 10
0.155
ff
ff
W
W
I W
IW
I
Answer
4
a. What is the spot size at the beam waist?
0.155 mm
b. What is the irradiance at the beam waist?
1.32 10 w/ 2
Answercm
265
Example, cont.Example, cont.
2 20
0 2
1/2 1/22 2
0 020 0
2 20
0
2( , ) exp
( ) ( )
( ) 1 1
(0.155)
WI z I
w z w z
z zw z W W
W z
Wz
c. What is the irradiance at the center of the beam
at 10 meters from the beam waist?
3
1/2 1/22 2
0 0
4
Answer2
154.6 mm0.488 10
( ) 10,0001 1 64.7
154.6
1.32(10 )(0,10 meters)
65
w z z
W z
I
23.15 w / cm
266
Gaussian Beam Profiling with Gaussian Beam Profiling with Knife EdgeKnife Edge
16%
84%
w(z)
Beam radius( )
2
w zx
( )
2
w zx
x
y
Knife edge
2
0
Knife Edge
1( , ) ( , ) ( , )
1 21
2 ( )
2( )
x
xt
p x z I y z dy I x z dx
xerf
w z
erf x e dt
Error Function Table
268
Geometrical Approach to Geometrical Approach to Gaussian Beam (Cont.)Gaussian Beam (Cont.)
• Standard Deviation (STD) beam radius (z): the radial distance of a circle that contains approximately 63.2% of the power (or irradiance drops down to 1/e 36.8%of the peak irradiance)
1/ 2 1/ 22 2
0 02
0 0
( ) 1 1z z
zk z
x
y
w(z)
(z)
86%
63.2%
1/e2
= 13.5%
1/e= 36.8%
-
269
Gaussian Beam Encircled EnergyGaussian Beam Encircled Energy• Encircled energy by the STD Beam radius (z):
it contains approximately 63.2% of the power
0
00
2
0
2
2
0
2
1( , ) ( , )2
21 exp
( )
1 exp( )
1( ( ), ) 1 63.2%
1.5* 2 * ( ), 99%
r
p r z I r z rdr
r
w z
r
z
p z ze
p z z
w0=1
(z)
x
y
r0
270
Spatial Filter (Cont)Spatial Filter (Cont)
• Spatial filter: A pinhole centered on the axis is placed at the Fourier transform plane (image plane) to block high-frequency noise
• The key is to choose the right pinhole size that blocks unwanted noise but maximally passes the laser’s energy
271
Spatial FilterSpatial Filter• Laser beam picks up intensity variation from scattering
by optical defects and particles in the air, known as spatial noise
• When a Gaussian beam is focused through a positive lens, equivalent to Fourier transform, the image at the focal plane will be mapped inversely proportional to the spatial frequencies
– Ideal Gaussian profile is imaged directly on axis
– Annulus of radius for noisy speckles:
nn dfr /
222
( )00 0 2
0
2( ) where
( )
r
w z
Actual Noise
WI r I e I
w z W
I I r I
0
ˆF
aW
272
Beam ExpanderBeam Expander
• Afocal system, or inverted telescope– Keplerian beam expander
• A microscope objective is often used for the first positive lens
– Galilean beam expander21_ ffLengthOverall
12 / ffionMagnificat
273
Spatial Filter (Cont)Spatial Filter (Cont)
• Relative power carried with a circle of radius r0
0 20
0 20
2
21( , ) ( , )2 1 exp
( )
1( ( ), ) 1 86%
(1.5 ( ), ) 99%
r
p r z I z dw z
p w z ze
p w z z
274
Beam Power & Beam RadiusBeam Power & Beam Radius
• Relative power through a spatial filter with a diameter of D:
/ 2
0
2
0
1( , ) ( , )2
11 exp
2
D
p D z I r z rdr
W D
f
0
3 ( ) ( , ) 99.3%opt opt
fD w z p D z
W Recommended :
275
Beam ImagingBeam Imaging
Spatial filter
Imaging lens
A small focused spot
276
Lens LayoutLens Layout
• The approaching wavefront refracts at the lens at different times, depending on its distance from the optical axes (r).
• The delays of the various regions of the wavefront are proportional to the thickness of the lens at each radial zone (r), as shown above for a spherical wave.
1ikA
U er
277
0
0
( ) ( ) ( ) (3.4)
( 1) ( )
k r k t t r nkt r
kt k n t r
where k(t0 – t(r)) is the phase delay caused by the free space region and it is assumed that the lens is surrounded by air (n = 1).
The wavefront’s velocity in the lens is slower than in air, so the section of the wavefront not in the glass will overtake the section that is in the glass.
278
0
0
2 0
1 0
2 1
( )
( )
The emerging wavefront is given by:
(3.5)
where the input wavefront was
(3.6)
so the output wavefront is
(3.7)
Since the phase change ( ) is a functionof thickness,
ik r
ik
ik r
U U e
U U e
U U e
k rt
1 2 3( ) ( ) ( ) (3.8)
, at a given zone of radius , thelens can be divided into three sections in orderto find the thickness, ( ). Therefore, thethickness as a function of zone radius, , is:
wher
t r t r t t r
r
t rr
2e is the "edge thickness" of the lens.t
279
Dividing the Thick Lens into Dividing the Thick Lens into Three SectionsThree Sections
280
x2 + y2 + z2 = R2 if @ coord center.Equation of a spherical surface in 3-Dfor center z = +R (rotational symetric system)
2 2 2 2
2 2
2 2 2
2 2
( )
define z S(r) r x
( ( ) )
( )
x y z R R
y
s r R R r
s r R R r
Circular symmetryuse negative sign (-) due to sign conventionfrom surface to r-axis is to left.
How much shift do we get to plane through vertex?
Z
r
R
s(r)
Spherical Surface
Center ofCurvature (CC)
Sag of Spherical Surface
281
2 2
1/ 22 2
1/ 22
2
324
22 4
2 2 3
( )
1
Since is small and / is squared,
1 can be expanded by a Taylor's series.
3
1 1 ...2 2 2! 2 3!
s r R R r
s r R R r
rs r R R
R
r r R
yyfRy R
R
Taylor Series/Maclaurin Series :
2 4 6
2 4 3 6
2 2
0
!
3( ) (1 ...
2 8 2 3!
We'll use the first two terms:
This is the lost distance( ) sag
by paraxial approx.....2 2
Where is:
1curvature
n n
n
y y ys y R R
R R R
y y Cs y
R
C
CR
Sag (Cont.)
282
1 3
1/ 22 21 10 1 1
1 10
1/ 22 23 30 2 2
( )
( )
( )
The thickness ( ) and ( ) are related to the sag of aspherical surface, which can be expressed as:
(3.9)
(3.10)
Rewriting the equations:
t r t r
t r t r
R R
R R
t r t R
t r t r
1/ 22
1 21
1/ 22
3 30 2 22
1 2
1 1
( ) 1 1
(3.11)
(3.12)
Assuming the lens radius ( ) is small compared to thesurface radii ( and ), a Taylor series approximationcan be made fo
r
R
rt r t R
R
rR R
2
11
2
2
10
3 30
( )2
( )2
r the square root parts of Equations 3.11and 3.12, and using the first two terms of that expansiongives:
(3.13)
(3.14)
rt r t
R
rt r t
R
283
2
01 2
2
0 01 2
2
2
2
01
0
Now Equation 3.8 becomes:
1 1( ) (3.15)
2
So the phase term, Equation 3.4 becomes:
1 1( ) 1 (3.16)
2
(3.17)
2
1 11
2
rt r t
R R
rk r kt k n t
R R
rknt k
f
rknt k n
R R
1 2
(3.18)*
Where we will define:
(3.19) 1 1 1
( 1)*
nf R R
284
WavefrontsWavefronts
2 1 2 1R R z z
2 1 2 1q q z z
285
Compare Gaussian toCompare Gaussian toSpherical Wave!Spherical Wave!
2 2
0tan
2 2
2
20
22 2 0
0 2
;
Radius of curvature is function of distance:
1
Waist as a function of distance is:
1
Due to these Gaussian properties
k kzi i iz i kz t i kz tR z r
R z Kr
E e e e E e e
zR z z
z
zW z W
z
02
, the distance alongrays are measured in imaginary terms;
. Follow Pedrotti 584-586.
1 1ori q z z iz
q z R z W z
complexradius of curvature
286
Lens Coupling Gaussian BeamLens Coupling Gaussian Beam
Lens system just like transferring spherical beam following 1st order optics
2 1
2 1
2 1
1 1 1
f
Lens(es) can be represented by a matrix:
y
q q
yA B
C D
f
287
Complex Radius of Curvature Complex Radius of Curvature of Gaussian Beamof Gaussian Beam
22
2
2 1 1 2 1 1
2 21 1
1 1
1 12 1
21
Radius of curvature of the beam is related to paraxialparameters:
So through a lens system, the radius of curvature ofa beam is:
1
yR
yR
y Ay B Cy D
yAR B CR D
AR By
CR D
1
11
1
1 12 2
1 1
1
;
AR B
CR D
AR B Aq BR q
CR D Cq D
289
Focus of a Gaussian Beam with LensFocus of a Gaussian Beam with Lens
2
1
21 0
2 2
2 220 0
2 2 2222
20
0
Complex radius of curvature of laser beam
1 1
Solve this by two boundary conditions:
@ Waist,
1
@ Coupling Mirror ( )
1 1 ( )
i
q R W
R
iq W
R
z WR z z
zz
Wz
a)
290
Solving:Solving:
220
2 22
220
9
26 2
0
13 40
4 140 13
40
20
1
22 2 2
2 0 20
02 1
00.7 1
632.8 10 0.7
21 7.09 10 0
0.7
1 5.03 10 0
1.85710 3.692 10
5.03 10
0 4.38 10 meter
0.9524
( ) [1 ] 5
WR z
z
W
W
W
W
W
Wq i i
zW z W
z
4.44(10 ) meter
291
1
2
2
3
3
1 0.7
0 1
1 0
1 1 11
2 1.5 1.5
1 0.004
0 1
1 0
11.5 1 1.5
0.64
1
0 1
1 0 1 01 1 0.004
0.5 0.5 20 1 0 11.5
0.64 3 3
R
R
A B
C D
b) Optical system transfer
1 0.7
0 1
1 0.53 0.7 0.63
0.53 0.63
Since complex radius of curvature, can't set 0 forsolving for .
B
292
9
2 24
1 0
11 2
1
2
1 632.8 101.0499
4.38 10
0.9524
1 0.53 0.952 0.7 0.63
0.53 0.9524 0.63
0.505 0.952 0.7 0.63
0.63 0.505
Multipy by c.c.
i i iq W
Aq Bq i q
Cq D
iq
i
i
i
2c) What is complex radius of curvature q ?
2 2 2
2
2
of denominator
0.318 0.6 0.44 0.3970.255 0.481 0.354 0.318
0.63 0.505
0.041 0.652 0.954
0.652
0.0613 1.463
ii i
q
iq
q i
293
Complex Radius of CurvatureComplex Radius of Curvature
22 2
2
9
2 22
7
22
1
but
6.32.8 10
2.01 10
q iR W
R
q iW
i
W
2d) q @beam waist of focused beam
294
-7
22
42
Real:
0.0613 0
0.0613m
Imaginary parts:
2.01 101.463
3.7 10 m beam waist
W
W
e) Setting real and imaginary part equal
295
Collimation of Gaussian BeamCollimation of Gaussian Beam
0
20
0
Lens for lasers are easier, no chromaticabberations
"Collimated distance" is between 2 adjacentRayleigh locations, or transverse size has
increased by 2 of waist 2
2 2
W
Wz
296
Focusing a Gaussian BeamFocusing a Gaussian Beam
2 1
2 1 21 2
1
2 1 21 1 2
12
111
201
1
1 01 1
10 1 0 11
f
1f f
11
f f
1f f
11
f f
As before in example problem:
@ waist,
z z
z z zz z
A B
z C D
z z zq z z
Aq Bq
zCq Cq
Wq i R
2
022 @ waist,
Solving for beam waist and distance in focusing space.
Wq i R
z
297
Focused Gaussian BeamFocused Gaussian Beam
22
0112 2 2
02 01
21
2 222 01
1
01 02
222201
0 1
020
1 1 11
f f
f ff
f
Focal length Raleigh rangeFocal length is small (large power)
f
2f
W
Wz
W W
zz
Wz
W W
WZ z
W
Assumptions :
From the above :
1
01
02
Recall if lens diameter 2
2F/#
2.44 F/# is larger for uniform irradianceacross lens
W
W
Recall diffraction :
298
Example:Example:
1 2
1 2
1 2
1 2
A HeNe 5 mW laser has a cavity of 34 cm lengthwith concave mirrors of radius 10 m 10m
632.8nm
0 1 Stable cavity condition
0.34 0.341 ; 1
10 10
R R
g g
g g
g g
1) Is the cavity stable/viable for Lasing?
1 2
0.066 0.066 0.004356; stable
1Center length symmetry
2
17 cmz z
2) Where is the beam waist?
299
Example, cont.Example, cont.
20
2 2
20
2
220
0
20
92 8
0
40
10 1
0.17
10 0.17 10.17
100.17 1 1.6711
0.17
1.29 m Raleigh range
1.29 632.8 1026 10
5.1 10 0.51 mm
zR z z
z
z
z
z
z
W
W
W
03) What is the spot size at the beam waist? W
300
Example, cont.Example, cont.
22 2
0 20
22
2
0 0ff2
0 00
9ff
1
170.51 1 0.2646
129
0.514 mm
)?
tan2
632.8 10
2 0.5
zW z W
z
W z
W W
z WW
ff
4) Determine the spot size on the mirrors?
5) Determine the beam full angular divergence (
3
4ff
ff
1 10
3.95 10 rad 0.395 mr2
0.79 mr