Rate Eq.

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


243

Laser Diodes –Electrical & Optical Properties

• Cavities, mode behavior

• Power – current plot

• Divergence

• Astigmatism

• Polarization

• Laser Diodes as geometrical light sources


244

Laser Diode Electronic Properties


245

Laser Diode Cavities


246

Laser Diode Power-Current Typical Curve


247

Laser Diode Mode Behavior


248

Laser Diode Polarization


249

Laser Diode Divergence


250

Laser Diode Divergence


251

Circularize by single-prism expansion


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


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


• 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


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


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


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


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


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


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

Rate Eq.

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