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Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification...

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Gaussian Beams Most lasers use cylindrically symmetric cavities Diffraction at cavity mirrors creates Gaussian Spherical Waves Recall E field for Gaussian R 2 y x Kr t i exp R U ) t , R , y , x ( u 2 2 0 R becomes the radius of curvature of the wave front These are really TEM 00 mode emissions from laser Creates a Gaussian shaped beam intensity 2 2 2 2 2 0 w r 2 exp w P 2 w r 2 exp I ) r ( I Where P = total power in the beam w = 1/e 2 beam radius w changes with distance z along the beam ie. w(z)
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Page 1: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Gaussian Beams Most lasers use cylindrically symmetric cavities Diffraction at cavity mirrors creates Gaussian Spherical Waves Recall E field for Gaussian

R2

yxKrtiexp

R

U)t,R,y,x(u

220

R becomes the radius of curvature of the wave front These are really TEM00 mode emissions from laser Creates a Gaussian shaped beam intensity

2

2

22

2

0 w

r2exp

w

P2

w

r2expI)r(I

Where P = total power in the beam w = 1/e2 beam radius w changes with distance z along the beam ie. w(z)

Page 2: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Measurements of Spotsize For Gaussian beam important factor is the “spotsize” Beam spotsize is measured in 3 possible ways 1/e radius of beam 1/e2 radius = w(z) of the radiance (light intensity) most common laser specification value 13% of peak power point point where emag field down by 1/e Full Width Half Maximum (FWHM) point where the laser power falls to half its initial value good for many interactions with materials useful relationship

erFWHM 1665.1

21177.1177.1e

rwFWHM

FWHMrwe

849.02

1

Page 3: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Gaussian Beam Changes with Distance The Gaussian beam radius of curvature with distance

220

z

w1z)z(R

Gaussian spot size with distance

21

2

20

0 w

z1w)z(w

Note: for lens systems lens diameter must be 3w0.= 99% of power Note: some books define w0 as the full width rather than half width As z becomes large relative to the beam asymptotically approaches

0

2

0

0)(w

z

w

zwzw

Asymptotically light cone angle (in radians) approaches

0wZ

zw

Page 4: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Rayleigh Range of Gaussian Beams Spread in beam is small when width increases < 2 Called the Rayleigh Range zR

2

0R

wz

Beam expands 2 for - zR to +zR from a focused spot Can rewrite Gaussian formulas using zR

2

2R

z

z1z)z(R

21

2R

2

0 z

z1w)z(w

Thus w = 2 w0 when z= 1.73 zr and peak power = 25% of z=0 Again for z >> zR

R0 z

zw)z(w

Page 5: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Beam Expanders Telescope beam expands changes both spotsize & Rayleigh Range Beam expanders are telescopes focused at infinity Kepler inverts the beam , Galilean does not For magnification m of side 2 relative side 1 then as before change of beam size is

0102 mww

Ralyeigh Range becomes

1R2

202

2R zmw

z

where the magnification is

1

2

f

fm

Page 6: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Example of Beam Divergence eg HeNe 4 mW laser has 0.8 mm rated diameter. What is its zR, spotsize at 1 m, 100 m and the expansion angle For HeNe wavelength = 632.8 nm Rayleigh Range is

m.x.

).(wzR 7940

103286

000407

220

At z = 1 metre

mm643.0m000643.0794.0

110004.0

z

z1w)z(w

21

2

221

2R

2

0

At z = 100 m >> zR

Radians10x04.50004.0

10x328.6

wz

)z(w 47

0

mm4.50m0504.0)10x04.5(100z)z(w 4

What if beam was run through a beam expander of m = 10

mmm.).(mww 4004000040100102

Radiansx.x.

mm5

4

1004510

10045

mm04.5m00504.0)10x04.5(100z)z(w 5

Hence get a smaller beam at 100 m by creating a larger beam first

Page 7: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Focused Laser Spot Lenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian Beams" App. Optics, pg. 658. v. 22, 5, 1983 Use the input beam waist distance as object distance s to primary principal point Output beam waist position as image distance s'' to secondary principal point

Page 8: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Gaussain Beam Lens Formulas Normal lens formula in regular and dimensionless form

111111

f

s

f

sor

fss

This formula applies to both input and output objects Gaussian beam lens formula for input beams includes Rayleigh Range effect

f

1

s

1

fs

zs

12R

in dimensionless form

1

f

s

1

1f

s

f

z

f

s

12

R

in far field as zR goes to 0 (ie spot small compared to lens) this reduces to geometric optics equations

Page 9: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Gaussain Beam Lens Behavior Plot shows 3 regions of interest for positive thin lens Real object and real image Real object and virtual image Virtual object and real image

Page 10: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Main Difference of Gaussian Beam Optics For Gaussian Beams there is a maximum and minimum image distance Maximum image not at s = f instead at

Rzfs

There is a common point in Gaussian beam expression at

1f

s

f

s

For positive lens when incident beam waist at front focus then emerging beam waist at back focus No minimum object-image separation for Gaussian Lens f appears to decrease as zR/f increases from zero i.e. Gaussian focal shift

Page 11: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Magnification and Output Beams Calculate zR and w0, s and s'' for each lens Magnification of beam

21

2R

20

0

f

z

f

s1

1

w

wm

Again the Rayleigh range changes with output

RR zmz 2

The Gaussian Beam lens formula is not symmetric From the output beam side

f

1

fs

zs

1

s

12

R

Page 12: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Special Solution to Gaussian Beam Two cases of particular interests Input Waist at First Principal Surface s = 0 condition, image distance and waist become

2

Rz

f1

fs

21

2

R

0

z

f1

w

f

w

Input Waist at First Focal Point s = f condition, image distance and waist become

fs

0w

fw

Page 13: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Gaussian Spots and Cavity Stability In laser cavities waist position is controlled by mirrors Recall the cavity g factors for cavity stability

ii r

L1g

Waist of cavity is given by

41

22121

21212

1

0gg2gg

gg1ggLw

where i=1=back mirror, i=2= front

Page 14: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Gaussian Waist within a Cavity Waist location relative to output mirror for cavity length L is

2121

212 gg2gg

Lg1gz

4

1

212

12

1

2

41

211

22

1

1 11

ggg

gLw

ggg

gLw

If g1 = g2 = g=0 (i.e. r = L) waist becomes

5.0z2

Lw 2

21

0

If g1 =0, g2 = 1 (curved back, plane front) waist is located z2 = 0 at the output mirror (common case for HeNe and many gas lasers) If g1 = g2 = 1 (i.e. plane mirrors) there is no waist

Page 15: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

General Laser Types Solid State Laser (solid rods): eg ruby Gas and Ion lasers: eg He-Ne, Argon Dye Lasers Semiconductor Laser: GaAs laser diode Chemical Lasers Free Electron Lasers

Page 16: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Solid State Lasers Was first type of laser (Ruby 1960) Uses a solid matrix or crystal carrier eg Glass or Sapphire Doped with ~1%-0.001% transition metal or rear earth ions eg Chromium (Cr) or Neodynmium (Nd) Mirrors at cavity ends (either on the rod or separate) Typically pumped with light Most common a Flash lamp: produces high power in ~1/1000 s Newer ones pumped by laser diodes (more efficient) Light adsorbed by doped ion, emitted as laser light Mostly operates in pulsed mode (newer CW)

Ted Maimin & laser

Original Ruby Laser

Page 17: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Flash Lamp Pumping Use low pressure flash tubes (like electronic flash) Xenon or Krypton gas at a few torr (mm of mercury pressure) Electrodes at each end of tube Charge a capacitor bank: 50 - 2000 F, 1-4 kV High Voltage pulse applied to tube Ionizes part of gas Makes tube conductive Capacitor discharges through tube Few millisec. pulse Inductor slows down discharge

Page 18: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Light Source Geometry Earlier spiral lamp: inefficient but easy Now use reflectors to even out light distribution For CW operation use steady light sources Tungsten Halogen or Mercury Vapour Use air or water cooling on flash lamps

Page 19: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Ruby Laser First laser built used Ruby rods: Maiman 1960 Crystal is Aluminium Oxide Al2O3: Sapphire 0.05% Cr3+ 3 level system: absorbs green/blue Emission at 694 nm (deep red) Pulsed operation

Page 20: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Ruby Laser Design Typically uses helix flash lamp Mirrors may be plated onto rod Mostly used for tattoo removal now Why Ruby laser was first – small rod inside of the helix flash Easy to get all light onto rod Ruby has strong meta-stable state (Maimin had measured it) Rod ends polished and optically aligned – simple & stable Mirrors aluminum plated on rods end (now aligned) Front mirror has small hole to allow output Much easier to create stable cavity & pump than gas lasers

First Laser opened up

Page 21: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Transition Metal Impurity Ion Energy levels Solid states using transition metals in clear matrix For Ruby Chromium Cr3+ ion Atom has energy levels (shells) (orbit)(shell)(no. electrons) 1s2 2s2 2p6 3s2 3p6 In ions unfilled orbital electrons interact Inter-electron coulomb interaction split the energies (capital letter the L quantum)(spin quantum) Ion then interacts with crystal field splits energy levels more

Page 22: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Rare Earth Impurity Ion Energy levels Spin of electrons interacts with orbit Splits the inter-electronic levels

Page 23: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Q Switch Pulsing Most solid states use Q switching to increase pulse power Block a cavity with controllable absorber or switch Acts like an optical switch During initial pumping flash pulse switch off Recall the Quality Factor of resonance circuits (eg RLC)

passlightperlostenergy

storedenergyQ

2

During initial pulse Q low Allows population inversion to increase without lasing No stimulated emission Then turn switch on Now sudden high stimulated emission Dump all energy into sudden pulse Get very high power level, but less energy Types of Q switches: Electro-Mechanical, electro-optics, acousto-optic, saturable dye

Page 24: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Q Switch Process During Laser Pulse Flash lamp rises to max then declines (~triangle pulse) Q switch makes cavity Q switch on after max pumping Low Q, so little spontaneous light Population inversion rises to saturation The Q switch creates cavity: population suddenly declines due to stimulated emission Laser pulse during high Q & above threshold conditions

Page 25: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Energy Loss due to Mirrors & Q Q switching can be related to the cavity losses Consider two mirrors with reflectance R1 and R2 Then the rate at which energy is lost is

rc

ERRE

211

where c = photon lifetime r = round trip time = 2L/c E = energy stored in the cavity Average number of photon round trips is the lifetime ratio

211

1

RRr

c

Page 26: Gaussian Beams - SFU.cagchapman/e894/e894l7p.pdfLenses focus Gaussian Beam to a Waist Modification of Lens formulas for Gaussian Beams From S.A. Self "Focusing of Spherical Gaussian

Q Equations for Optical Cavity Rewrite energy equation in terms of photon lifetime c First note the energy lost in the time of one light cycle tf = 1/f

cc

fcycle/lost f

EEtE

where f = frequency Thus the cavity's Q is

c

c

cycle/lost

f

f

E

E

E

EQ

2

22

Thus for a laser cavity:

212121

rc RR1

L4

RR1c

fL4

RR1

f2f2Q

Q switch: go form high reflectivity to low reflectivity on one mirror Also Q is related to the bandwidth of the laser (from resonance cavity circuits).

f

fQ

Thus lifetime relates to the bandwidth

c

f

2

1


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