Matrix Methods in Optics For more complicated systems use Matrix methods & CAD tools Both are based on Ray Tracing concepts Solve the optical system

Transcription

1 Matrix Method in Optic For more complicated ytem ue Matrix method & tool oth are baed on Ray Tracin concept Solve the optical ytem by tracin may optical ray In ree pace a ray ha poition and anle o direction y i radial ditance rom optical axi V i the anle (in radian) o the ray No aume you ant to a Tranlation: ind the poition at a ditance t urther on Then the baic Ray equation are in ree pace makin the parallex aumption y y Vt V V

2 Matrix Method: Tranlation Matrix an deine a matrix method to obtain the reult or any optical proce onider a imple tranlation ditance t Then the Tranlation Matrix (or T matrix) V y t V y V y The revere direction ue the invere matrix V y t V y V y V y

3 General Matrix or Optical evice Optical urace hoever ill chane anle or location Example a len ill keep ame location but dierent anle Reerence or more len matrice & operation. Gerrard & J.M. urch, Introduction to Matrix Method in Optic, over 994 Matrix method equal Ray Trace Proram or imple calculation

4 General Optical Matrix Operation Place Matrix on the let or operation on the riht an olve or calculate a inle matrix or the ytem [ ][ ][ ] V y M M M V y object len imae V y V y

5 Solvin or imae ith Optical Matrix Operation For any len ytem can create an equivalent matrix ombine the len (mirror) and pacin beteen them reate a inle matrix [ ] [ ][ ] [ ] M M M M ytem n L No add the object and imae ditance tranlation matrice [ ][ ][ ] object len imae M M M ( ) Imae ditance i ound by olvin or Imae maniication i m

6 Example Solvin or the Optical Matrix To len ytem: olve or imae poition and ie iconvex len 8 cm located 4 cm rom 3 cm tall object Second len biconcave - cm located d6 cm rom irt len Then the matrix olution i X X.4.5 X X Solvin or the imae poition uin the matrix & X matrix: cm X or X Then the maniication i m Thu the object i at cm rom nd len, -3 cm hih

7 Matrix Method and Spread Sheet Eay to ue matrix method in Excel or matlab or maple Ue mmult array unction in excel Select array output cell (e. matrix) and enter mmult( Select pace cell then comma Select len cell (e mmult(g5:h6,i5:j6) ) Then do controlhitenter (very important) Here i example rom previou pae

8 Optical Matrix Equivalent Len For any len ytem can create an equivalent matrix & len ombine all the matrice or the len and pace The or the combined matrix here RP irt len let vertex RP lat len riht mot vertex n index o reraction beore t len n index o reraction ater lat len

9 Example ombined Optical Matrix Uin To len ytem rom beore iconvex len 8 cm Second len biconcave - cm located 6 cm rom Then the ytem matrix i Second ocal lenth (relative to H ) i cm Second ocal point, relative to RP (econd vertex). 5 rp. 4 cm. 4 Second principal point, relative to RP (econd vertex). 5 H cm

10 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 iω 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 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

11 Gauian eam ume a Gauian haped beam I( r ) r exp P π r exp I Where P total poer in the beam /e beam radiu at point ()

12 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.665r FWHM.77.77r r e.849 e FWHM e

13 Gauian eam hane ith itance The Gauian beam radiu o curvature ith ditance ) R( λ π Gauian pot ie ith ditance ) ( π λ Note: or len ytem len diameter mut be 3. 99% o poer Note: ome book deine a the ull idth rather than hal idth become lare relative to the beam aymptotically approache ) ( π λ π λ ymptotically liht cone anle (in radian) approache ( ) Z π λ θ

14 Rayleih Rane o Gauian eam Spread in beam i mall hen idth increae < alled the Rayleih Rane R R π eam expand or - R to R rom a ocued pot an rerite Gauian ormula uin R R ( ) λ R ( ) R ain or >> R ( ) R

15 eam 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 λ R here the maniication i m

16 Example o eam iverence 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 λ 63.8 nm Rayleih Rane i t metre π λ π(.4) 6.38x R m ( ).643 m R mm t m >> R θ ( ) 7 4 ( ) λ π 6.38x π.4 θ ( 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 θ ( ) m θ m. 4x θ ( 5.4x 5. 4x 5 Radian ).54 m 5.4 mm Hence et a maller beam at m by creatin a larer beam irt

17 Focued Laer Spot Lene ocu Gauian eam to a Wait Modiication o Len ormula or Gauian eam From S.. Sel "Focuin o Spherical Gauian eam" pp. 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

18 Gauain eam 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

19 Gauain eam Len ehavior 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

20 Main ierence o Gauian eam Optic For Gauian eam 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

21 Maniication and Output eam alculate R and, and '' or each len Maniication o beam R m ain the Rayleih rane chane ith output R R m The Gauian eam len ormula i not ymmetric From the output beam ide ( ) R

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

23 Gauian Spot and avity 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 λ here iback mirror, i ront

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