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

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1 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 ω anular requency π U max value o E ield R radiu rom ource t time K propaation vector in direction o motion r unite radial vector rom ource x,y plane poition perpendicular to R A R increae ave become Gauian in phae R become the radiu o curvature o the ave ront Thee are really TEM mode emiion rom laer

2 Gauian Beam Aume a Gauian haped beam I( r ) I r exp P π r exp Where P total poer in the beam /e beam radiu at point ()

3 Meaurement o Spotie beam pot ie i meaured in 3 poible ay /e radiu o beam /e radiu () o the radiance (liht intenity) mot common laer peciication value 3% o peak poer point point here ema ield don by /e 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.386r e FWHM.693r e

4 Gauian Beam Chane ith Ditance The Gauian beam radiu o curvature ith ditance ) R( π Gauian pot ie ith ditance ) ( π 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 ) ( π Aymptotically liht cone anle (in radian) approache Z ) ( π θ

5 Rayleih Rane o Gauian Beam Spread in beam i mall hen idth increae < Called the Rayleih Rane R R π Beam expand or - R to R rom a ocued pot Can rerite Gauian ormula uin R R ( ) R ( ) R Aain or >> R ( ) R

6 Beam Expander Telecope beam expand chane both potie and Rayleih Rane For maniication m o ide relative ide then a beore chane o beam ie i Ralyeih Rane become m π R m R here the maniication i m

7 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 π π(.4) 6.38x R m ( ).643 m R mm At m >> R θ 7 4 ( ) π 6.38x π.4 5.4x Radian ( ) θ ( 5.4x 4 ).54 m 5.4 mm What i beam a run throuh a beam expander o m m (. 4 ). 4 m 4 mm θ m ( ) θ θ m. 4x ( 5.4x 5. 4x 5 Radian ).54 m 5.4 mm Hence et a maller beam at m by creatin a larer beam irt

8 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

9 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

10 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

11 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

12 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

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

14 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 π here iback mirror, i ront

15 Gauian Wait ithin a Cavity Wait location relative to output mirror or cavity lenth L i ( ) L [ ] [ ] 4 4 L L π π I (i.e. r L) ait become.5 L π 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

16 Diraction Liht ave pa throuh an aperture Huyen' Principal: each point on averont act a ource o another ave I liht comin rom ininity point ource at ininity or parallel beam (laer) Liht at lit ede diract Intererence eect beteen the ave at each point chaned

17 Frenel and Fraunhoer Intererence Both aume liht ource at ininity (near parallel liht - laer beam) Frenel intererence Pattern created near the diraction point Much more complex equation Fraunhoer Intererence Diracted liht ened at ininity I ocu lit ith len ame a at ininity Mot common eect

18 Fraunhoer Intererence For inle lit idth b Intenity ollo the pattern o a ynch unction here I ( β ) β I in( β β π b in( θ ) ) θ anular deviation o pattern rom minimum

19 Zero are at here N i any inteer Fraunhoer Intererence Pattern β ± Nπ Lare d little pattern, mall d pattern pread out

20 Circular Fraunhoer Intererence Intererence chane or circular openin mot important or laer ytem Called an Airy Dik For inle circular aperture diameter D Intenity ollo the pattern here I ( β ) β I J ( β β ) π D in( θ ) J Beel unction o irt kind, order θ anular deviation o pattern rom minimum

21 Comparion o Circular and Slit Fraunhoer Intererence Slit produce maller idth pattern

22 Minimum Spotie o Focued Laer Beam For beam much maller than len limited by ait ie o input beam Hence i ait in et at the ocu then π NOTE: len aberration may modiy thi by a actor e He-Ne laer i ocued throuh a len ith 7 mm 63.8 nm out.4 mm What i the minimum pot produced? Aume input ait i at ocu π x (.7 ) 3.5x 4 π 4x For inlet len multiply thi by.333 or 4.7 micron 3.5 µ m

23 Diraction Limited pot I laer beam ill the len then diraction limited Openin o idth D Minimum pot i to point o irt ero in diraction I b in( θ ) d min D bd Since circular eectively Airy diraction increaed by a actor o. d min.44 D min

24 Depth o Focu Spot i in ocu i ait expand le than 5% Uin ait ormula ( ) π Manipulatin thi uin a binomial expanion then at 5% chane then ±.3π e previou He-Ne laer i ocued throuh a len ith 7 mm 63.8 nm 3.5 micron hat i the depth o ocu o pot produced? ( 6 ) 3.5x 5.97 x m 9.7 m.3π ± x µ

25 Diraction Limit and Gauian Optic Both ive lihtly dierent aner becaue lookin at dierent beam part e He-Ne laer i ocued throuh a len ith 5 mm 63.8 nm D mm What i the minimum pot produced? I ant irt minimum o Airy dik then d.44 D.44(.5 )( 6.38x. ) 7.7 X 7 6 min m 7.7µ I aume input ait i at ocu uin rane ormula Let pot be typical /3 len o ait out 3.3 mm π x (.5 ) 3.5x π µ m Dierence come rom minimum diameter o Airy dik vere /e radiu m