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

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Cornelius Griffin
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1 Gauian Beam Mot laer ue cylindrically ymmetric cavitie Diraction occur at ede o circular cavity mirror Create Gauian Spherical Wave Recall E ield or Gauian U x y u( x, y,r,t ) exp i t Kr R R R become the radiu o curvature o the ave ront Thee are really TEM mode emiion rom laer Create a Gauian haped beam intenity I( r ) r exp P r exp I Where P = total poer in the beam = /e beam radiu chane ith ditance alon the beam ie. ()

2 Meaurement o Spotie For Gauian beam important actor i the potie Beam potie i meaured in 3 poible ay (a) /e radiu o beam (b) /e radiu = () o the radiance (liht intenity) mot common laer peciication value 3% o peak poer point point here ema ield i don by /e (c) Full Width Hal Maximum (FWHM) point here the laer poer all to hal it initial value ood or many interaction ith material Ueul relationhip FWHM.77.77r r e.849 FWHM FWHM.665r e e

3 Gauian Beam Chane ith Ditance The Gauian beam radiu o curvature o E ield ith ditance ) R( Gauian pot ie ith ditance ) ( Note: or len ytem len diameter mut be 3.= 99% o poer Note: ome book deine a the ull idth rather than hal idth A become lare relative to the beam aymptotically approache ) ( Aymptotically liht cone anle (in radian) approache Z

+ R rom a ocued pot Can rerite Gauian ormula uin R ( R ( ) )

4 Rayleih Rane o Gauian Beam Spread in beam i mall hen idth increae < Called the Rayleih Rane R R Beam expand or - R to + R rom a ocued pot Can rerite Gauian ormula uin R ( R ( ) ) R Thu = hen =.73 r and peak poer = 5% o = Aain or >> R ( ) R R

5 Beam Expander Telecope beam expand chane both potie & Rayleih Rane Beam expander are telecope ocued at ininity Kepler invert the beam, Galilean doe not For maniication m o ide relative ide then a beore chane o beam ie i Ralyeih Rane become R here the maniication i m m m R

6 Example o Beam Diverence e HeNe 4 mw laer ha.8 mm rated diameter. What i it R, potie at m, m and the expanion anle For HeNe avelenth = 63.8 nm Rayleih Rane i At = metre (. 4 ) R. 794 m x ( ).643 m R mm At = m >> R ( ) 7 4 ( ) 6.38x.4 ( 5.4x 4 5.4x ).54 m Radian 5.4 mm What i beam a run throuh a beam expander o m = m (. 4 ). 4 m 4 mm ( ) m m x ( 5.4 x 5. 4x 5 Radian ).54 m 5.4 mm Hence et a maller beam at m by creatin a larer beam irt

Focued Laer Spot Lene ocu Gauian Beam to a Wait Modiication o Len ormula or Gauian Beam From S.A. Sel "Focuin o Spherical Gauian Beam" App. Optic, p. 658.

7 Focued Laer Spot Lene ocu Gauian Beam to a Wait Modiication o Len ormula or Gauian Beam From S.A. Sel "Focuin o Spherical Gauian Beam" App. Optic, p v., 5, 983 Ue the input beam ait ditance a object ditance to primary principal point Output beam ait poition a imae ditance '' to econdary principal point

8 Gauain Beam Len Formula Normal len ormula in reular and dimenionle orm or Thi ormula applie to both input and output object Gauian beam len ormula or input beam include Rayleih Rane eect R in dimenionle orm R in ar ield a R oe to (ie pot mall compared to len) thi reduce to eometric optic equation

9 Gauain Beam Len Behavior Plot ho 3 reion o interet or poitive thin len Real object and real imae Real object and virtual imae Virtual object and real imae

Main Dierence o Gauian Beam Optic For Gauian Beam there i a maximum and minimum imae ditance Maximum imae not at = intead at There i a common point in Gauian beam expreion at R For

10 Main Dierence o Gauian Beam Optic For Gauian Beam there i a maximum and minimum imae ditance Maximum imae not at = intead at There i a common point in Gauian beam expreion at R For poitive len hen incident beam ait at ront ocu then emerin beam ait at back ocu No minimum object-imae eparation or Gauian Len appear to decreae a R / increae rom ero i.e. Gauian ocal hit

11 Maniication and Output Beam Calculate R and, and '' or each len Maniication o beam R m Aain the Rayleih rane chane ith output R R m The Gauian Beam len ormula i not ymmetric From the output beam ide R

12 Special Solution to Gauian Beam To cae o particular interet Input Wait at Firt Principal Surace = condition, imae ditance and ait become R R Input Wait at Firt Focal Point = condition, imae ditance and ait become

13 Gauian Spot and Cavity Stability In laer cavitie ait poition i controlled by mirror Recall the cavity actor or cavity tability i i r L Wait o cavity i iven by 4 L here i==back mirror, i== ront

14 Gauian Wait ithin a Cavity Wait location relative to output mirror or cavity lenth L i L 4 4 L L I = = = (i.e. r = L) ait become.5 L I =, = (curved back, plane ront) ait i located = at the output mirror (common cae or HeNe and many a laer) I = = (i.e. plane mirror) there i no ait

15 General Laer Type Solid State Laer (olid rod): e ruby Ga and Ion laer: e He-Ne, Aron Dye Laer Semiconductor Laer: GaA laer diode Chemical Laer Free Electron Laer

Solid State Laer Wa irt type o laer (Ruby 96) Ue a olid

% tranition metal or rear earth ion e Chromium (Cr) or

eparate) Typically pumped ith liht Mot common a Flah lamp:

eicient) Liht adorbed by doped ion, emitted a laer liht

16 Solid State Laer Wa irt type o laer (Ruby 96) Ue a olid matrix or crytal carrier e Gla or Sapphire Doped ith ~%-.% tranition metal or rear earth ion e Chromium (Cr) or Neodynmium (Nd) Mirror at cavity end (either on the rod or eparate) Typically pumped ith liht Mot common a Flah lamp: produce hih poer in ~/ Neer one pumped by laer diode (more eicient) Liht adorbed by doped ion, emitted a laer liht Motly operate in puled mode (neer CW) Ted Maimin & laer Oriinal Ruby Laer

capacitor bank: 5 - F, -4 kv Hih Voltae pule applied to tube Ionie part o a

17 Flah Lamp Pumpin Ue lo preure lah tube (like electronic lah) Xenon or Krypton a at a e torr (mm o mercury preure) Electrode at each end o tube Chare a capacitor bank: 5 - F, -4 kv Hih Voltae pule applied to tube Ionie part o a Make tube conductive Capacitor dichare throuh tube Fe milliec. pule Inductor lo don dichare

18 Liht Source Geometry Earlier piral lamp: ineicient but eay No ue relector to even out liht ditribution For CW operation ue teady liht ource Tunten Haloen or Mercury Vapour Ue air or ater coolin on lah lamp

19 Ruby Laer Firt laer built ued Ruby rod: Maiman 96 Crytal i Aluminium Oxide Al O 3 : Sapphire.5% Cr 3+ 3 level ytem: aborb reen/blue Emiion at 694 nm (deep red) Puled operation

Ruby Laer Dein Typically ue helix lah lamp

tattoo removal no Why Ruby laer a irt mall

onto rod Ruby ha tron meta-table tate (Maimin

alined imple & table Mirror aluminum plated

hole to allo output (Make R<) Much eaier to

20 Ruby Laer Dein Typically ue helix lah lamp Mirror may be plated onto rod Motly ued or tattoo removal no Why Ruby laer a irt mall rod inide o the helix lah Eay to et all liht onto rod Ruby ha tron meta-table tate (Maimin had meaured it) Rod end polihed optically alined imple & table Mirror aluminum plated on rod end (no alined) Front mirror ha mall hole to allo output (Make R<) Much eaier to create table cavity & pump than a laer Firt Laer opened up

21 Tranition Metal Impurity Ion Enery level Solid tate uin tranition metal in clear matrix For Ruby Chromium Cr 3+ ion Atom ha enery level (hell) (orbit)(hell) (no. electron) p 6 3 3p 6 In ion unilled orbital electron interact Inter-electron coulomb interaction plit the enerie (capital letter the L quantum) (pin quantum) Ion then interact ith crytal ield plit enery level more

22 Rare Earth Impurity Ion Enery level Spin o electron interact ith orbit Split the inter-electronic level

23 Q Sitch Pulin Mot olid tate ue Q itchin to increae pule poer Block a cavity ith controllable aborber or itch Act like an optical itch Durin initial pumpin lah pule itch o Recall the Quality Factor o reonance circuit (e RLC) Q enery tored enery lot per liht pa Durin initial pule Q lo Allo population inverion to increae ithout lain No timulated emiion Then turn itch on No udden hih timulated emiion Dump all enery into udden pule Get very hih poer level, but le enery Type o Q itche: Electro-Mechanical, electro-optic, acouto-optic, aturable dye

24 Q Sitch Proce Durin Laer Pule Flah lamp rie to max then decline (~trianle pule) Q itch make cavity Q itch on ater max pumpin Lo Q, o little pontaneou liht Population inverion rie to aturation The Q itch create cavity: population uddenly decline due to timulated emiion Laer pule durin hih Q & above threhold condition

25 Enery Lo due to Mirror & Q Q itchin can be related to the cavity loe Conider to mirror ith relectance R and R Then the rate at hich enery i lot i E c RR here c = photon lietime r = round trip time = L/c E = enery tored in the cavity Averae number o photon round trip i the lietime ratio c r r R R E

26 Q Equation or Optical Cavity Rerite enery equation in term o photon lietime c Firt note the enery lot in the time o one liht cycle t = / E here = requency Thu the cavity' Q i Q Thu or a laer cavity: Q c lot / cycle E E lot / cycle Et c E E c E c R R c R R R R r 4 L c 4 L Q itch: o orm hih relectivity to lo relectivity on one mirror Alo Q i related to the bandidth o the laer (rom reonance cavity circuit). Q Thu lietime relate to the bandidth c

Gaussian Plane Waves Plane waves have flat emag field in x,y Tend to get distorted by diffraction into spherical plane waves and Gaussian Spherical

$Gaussian Plane Waves Plane waves have flat emag field in x,y Tend to get distorted by diffraction into spherical plane waves and Gaussian Spherical$ Gauian Plane Wave Plane ave have lat ema ield in x,y Tend to et ditorted by diraction into pherical plane ave and Gauian Spherical Wave E ield intenity ollo: U ( ) x y u( x, y,r,t ) exp i ω t Kr R R here