Theory study about quarter-wave-stack dielectric mirrors

Transcription

1 Theor tud about quarter-wave-tack delectrc rror

2 Stratfed edu tratted reflected reflected Stratfed edu tratted cdet cdet T T Frt, coder a wave roagato a tratfed edu. A we kow, a arbtrarl olared lae wave could be reolved to two wave, oe wth t electrc vector eredcular to the lae of cdece, we hall eak of a travere electrc wave deoted b T, aother oe wth t agetc vector eredcular to the lae of cdece we hall eak of a travere agetc wave deoted b T. ere, we take the lae of cdece to be,-lae, beg the drecto of tratfcato.

3 3 So, ow for a T wave,, awell equato reduce to: c c c c µ µ µ are equato of ad ol. B elatg we have:, l µ µ c k d d k

4 Defe:, Y U the, the above equato ca be wrtte a: k Y d Y d U d U d dl µ d U du d ow, the ter o the left a fucto of ol whlt the ter o the rght deed ol o. ece th equato ol hold f each de equal to a cotat -K a: Y d Y d K It wll be coveet to et The we get the oluto of Y: Coequetl of the for, d U d dl µ d K k where U a fucto of. Fro equato of lat age, we ee that gve b ereo of the ae for: a du d k a [ co ta t] e Y U e V e k k a t a t k U K U 4

5 It had bee derved Bor ad Wolf «Prcle of Otc» that: U V U V ere, a trao atr ad th atr uodular,. The gfcace of to relate the - ad -cooet of the electrc or agetc vector the lae to the cooet a arbtrar lae cotat. I ked the colcated dervato here. ow, we aw that the kowledge of U ad V uffcet for the colete ecfcato of the feld. ece for the uroe of deterg the roagato of a lae wave through a tratfed edu, the edu ol eed be ecfed b a arorate uodular atr. So, we hall call the charactertc atr of the tratfed edu. 5

6 Stratfed edu A T wave: reflected ε R µ θ tratted T θ cdet A Let A, R, T deote a before the altude obl cole of the electrc vector of the cdet, reflected, ad tratted wave. Further, let, µ,, µ, deote, a the above cture how. Ad the boudar codto dead that the tagetal cooet of ad hall be cotuou acro each of the two boudare of the tratfed edu. 6

7 7 If the relato alo ued, we get the followg relato for a T wave: Aue the charactertc atr to be: o, S r r r µ T V T U R A V R A U co co µ µ V U V U T T R A R A co co µ * µ *

8 Fro the above equato, we ca obta the reflecto ad trao coeffcet of the fl: r t R A T A µ co I ter of r ad t, the reflectvt ad travt are R r T, t The hae δ r of r a be called the hae chage o reflecto ad the hae of the hae chage o trao. So, the ot ow to get the charactertc atr of the tratfed edu, the we ca obta reflecto ad trao coeffcet. Coequetl, we obta the reflectvt ad travt ad hae hft troduced b reflecto ad trao. 8

9 ow, coder a hoogeeou delectrc fl. I th cae, µ, µ are cotat. Aue the eda to be oagetc µ. θ deote the agle whch the oral to the wave ake wth the -a the fl. A how «Prcle of Otc», a hoogeeou ha the followg charactertc atr: co co k co co co co µ Th charactertc atr decrbe oe delectrc fl. For a ultlaer delectrc rror, we coder a tack of ar alterate laer wth refractve dce ad ad wth reectve thcke d ad d, deoted uo a ubtrate wth refractve de. The ae, we aue all the eda to be oagetc µ. 9

10 d d d d Subtrate S ar ar ar For th tructure, the charactertc atre correodg to ad are: co co co co d co co co d co

11 Let be the charactertc atr correodg to oe erod: erod: co co co co co co co aue to be co Oce we get the atr, we ca obta the reflecto ad trao coeffcet b: t r I geeral codto, t ver colcated to calculate a th ower of a atr. co co

12 A quarter-wave-tack ultlaer delectrc rror I artcular cae, the alterate laer could be fabrcated wth a arorate thcke correodg to ecfed wavelegth ad cdet agle to let the otcal ath legth equal to quarter-wavelegth. The, t ha: So, the charactertc atre correodg to ad are reduced to: 4 co co d d co co

13 3 The reflecto ad trao coeffcet could be how a: t r

14 4 Aue the lght to be oral cdece: The reflecto ad trao coeffcet could be reduced a: The reflectvt ad travt are: Wth the creae of, ad f < the,. So, tack quarter-wave ultlaer delectrc a falar ethod to fabrcate a ver hgh reflectvt rror. co, co o t r, t r T R

15 ow, go back to the equato of reflecto ad trao coeffcet. Obvoul, the reflecto ad trao coeffcet are real uber. It ea that uder deal codto, o fabrcato error, o aborto, o wavelegth devato, there wll ot be hae hft troduced b reflecto ad trao fro a quarter-wave-tack delectrc rror for a T wave. Ad t ca be derved eal for a T wave that the reflecto ad trao coeffcet are real uber, too. So, o hae hft for both cooet of a olared lght ea o chage for olarato tatu. Awa, we hould coder oe error factor a we ca. 5

16 We got oe bac forato for the fabrcato ad error of the rror fro the coa. But, the ca ot offer uch ore detaled decrto about the refracto de ad thcke of the delectrc laer due to the rotecto of techolog. I geeral, a chee of 37 laer Ta O 5 /SO wa aled. Ue the refracto dce lke:.4 Ta SO O rror of ad :.---. rror of thcke:.3 totall for 37 laer Laer wavelegth devato: Δλ a <.5 fro laer tet reort 6

17 ow, for a oral cdece, coder the wort codto each error reache t bgget value for a T wave. The, d co d get:.7 rad a a Aue δ.rad, the charactertc atr of laer are: co... co. co... co [ ]

18 So, we ca get the reflecto coeffcet a: r r arg r R r o.96 o It how above that the T wave wll have a coo 8 degree reflecto hft ad a addtoal about.96 degree hae hft troduced b all error of the rror fabrcato ad laer wavelegth devato for oe te reflecto fro the quarter-wave-tack delectrc rror. 8

19 For a T wave, the ae atr hold, wth relaced b q: co q q co co q q co q µ o µ,, µ co q co... co. co... co [ ]

20 r r arg r.96 o R r Coare the reult of T a T we could fd out that the hae hft troduced b all error of the rror fabrcato ad laer wavelegth devato for oe te reflecto wll be ae. It ea the olarato tatu wll be ataed ae hae hft for T ad T wave ad ol the olarato drecto wll chage T wave ha a 8 degree hae hft but T ha ot. otce: Ol uder the codto of oral cdece, the T ad T wave wll have the ae hae hft troduced b all error of the rror fabrcato ad laer wavelegth devato

21 Three ot had bee aued for th calculato:. o aborto whe laer a through rror. Could we gore t?. The ultlaer delectrc laer ateral otroc. I t? eed certfed 3. Aroatel decrbe the gaua bea a a lae wave. ow to deal wth a real gaua bea?