Power Transmittance of a Laterally Shifted Gaussian Beam through a Circular Aperture

Transcription

1 Poer Trnsmittnce of Lterlly Shifted Gussin Bem through Circulr Aperture Triq Shmim Khj 1 nd Syed Azer Rez 1 1. Deprtment of Electricl Engineering, Lhore University of Mngement Sciences, DHA, Lhore 5479, Pkistn Abstrct Gussin bems re often used in opticl systems. The fundmentl Gussin TEM mode is the most common of the Gussin modes present in vrious opticl devices, systems nd equipment. Within n opticl system, it is common tht this Gussin TEM bem psses through circulr perture of finite dimeter. Such circulr pertures include irises, sptil filters, circulr Photo-Detectors (PDs) nd opticl mounts ith circulr rims. The mgnitude of opticl poer pssing through finite-sized circulr perture is ell-documented for cses here the Gussin bem psses through the center of the cler circulr perture, nd is chopped off symmetriclly in ll rdil directions on given plne. More often thn not, non-xil incident Gussin Bem is not blocked in rdilly uniform mnner by circulr perture. Such situtions rise due to lterl displcement of the bem from tilted glss blocks, mnufcturing errors nd imperfect surfce fltness or prllelness of surfces. The frction of opticl poer of lterlly-shifted Gussin Bem pssing through circulr perture is clculted in this pper through conventionl integrtion techniques. 1. Introduction In most lser-bsed opticl systems, bems Lser Source (LS) pss through some opticl components ith circulr pertures such s sphericl lenses, circulr mirrors, irises nd circulr sptil filters, circulr grtings, opticl velength filters, ttenutors nd mny more. Moreover, Photo-Detectors (PDs), used for detection of the incident opticl poer, commonly hve circulr ctive re. Typiclly the vrious opticl elements inside n opticl system re ligned so tht the opticl xis of the system is coincident ith the opticl xis of individul elements such s lenses nd curved mirrors hich llos the lser bem to

2 propgte through the center of these elements. In this cse, ll elements in the opticl system re rottionlly symmetric b the opticl xis of the system. Most lser sources such s He-Ne lsers emit the fundmentl Gussin Mode TEM. When Gussin Bem propgtes through opticl elements ith circulr pertures, frction of the totl bem poer is blocked due to finite cler perture rdius of ech element. In the cse of PD ith circulr ctive re, only the frction of opticl poer hich is incident on the PD ctive re contributes to the generted photocurrent hile the remining opticl poer is not recorded. The trnsmitted poer of Gussin bem pssing through circulr perture is ell-documented for the cse hen the center of the bem irrdince profile coincides ith the center of the perture [1]. In such cse, the to-dimensionl irrdince profile function of the Gussin Bem is integrted over the dimensions of the cler perture of the circulr opticl element nd multiplied, depending on bem polriztion, by the Fresnel Poer Trnsmission Coefficient [] to determine the poer trnsmitted. As the centers of the bem profile nd perture coincide, the integrtion is rther simple due to rdilly symmetric integrl. On the other hnd, the clcultion of trnsmitted opticl poer of Gussin Bem becomes tedious hen the bem is lterlly displced ith respect to the center of cler perture of n opticl element. A simpler cse of poer trnsmission or detection of lterlly-shifted bem through squre perture hs been presented in [3]. In the cse of circulr perture, the integrtion of the to-dimensionl Gussin irrdince function over circulr cler perture is not rdilly symmetric hich renders it non-trivil integrl to solve. The effect on trnsmitted opticl poer due to lterl bem shift is negligible hen the cler perture dimeter is much lrger thn the Full Width Hlf of Mximum (FWHM) bem dimeter s ell s the mgnitude of lterl shift. But trnsmitted poer cn be severely ffected by smll lterl shifts if the perture dimeter is comprble to the FWHM bem dimeter. This is often the cse hen bems re

3 pssing through smll irises or sptil filters or if bems re incident on PDs ith smll ctive re dimeters. Lterl displcement of Gussin bem is very common ithin opticl setups here it cn be result of lterl or tilt mislignments or defects in opticl components. Also such lterl bem displcement cn be the result of pltes hich re often designed s edge-shped to mitigte bck-reflection effects. It is lso knon tht tunble focus lenses often exhibit non-prllelism of to surfces tht hold liquid beteen them, resulting in n ngulr devition of the bem hich psses through them [4]. This ngulr devition results in lterl bem shift t ny plne fter the lens. In this pper, e clculte the percentge trnsmitted opticl poer of bem hich is lterlly shifted ith respect to the center of circulr perture. In the cse of photo-detection ith circulr detectors, this ill signify the percentge of incident opticl poer hich is recorded by PD.. Trnsmittnce of Non-Centered Gussin Bem through Circulr Aperture Fig.1 shos Gussin Bem hich is incident such tht its pek of irrdince profile is coincident ith the center of circulr perture. We lso sho in Fig.1, lterlly displced Gussin Bem ith respect to the center of circulr cler perture. In this section, e im to determine the trnsmitted opticl poer in the ltter cse. Fig.1 Gussin Bem incidence on-center nd off-center of circulr perture.

4 The irrdince profile, I(x, y, z), in Crtesin Coordintes (x,y,z), of Gussin bem ith minimum bem ist t z =, is given by: I(x, y, z) = I Pek ( (z) ) exp ( ((x ) + (y ) ) ) (z) If the plne of the circulr perture is locted t z = z, then (z ) = is the 1/e bem ist rdius nd I Pek is the pek irrdince t the center of the Gussin Bem t this loction. From Eq.(1), the irrdince profile of the Gussin bem t z = z is given by: (1) I(x, y, z) = I Pek ( ) exp ( ((x ) + (y ) ) ) As shon in Fig.1, e choose the coordintes (x,y) ccording to the circulr perture such tht the point (,) signifies the center of the perture. The lterl displcement of the center of the Gussin Bem is d. As the choice of primry coordinte xes is rbitrry, the direction of bem displcement from the center loction is tken to be the x-xis. The irrdince profile in Eq.() is then expressed in (x,y) coordintes s: () I(x, y, z) = I Pek ( ) exp ( ((x d) + y ) ) = I ( ) exp ( (x + y + d xd) ) Which cn be simplified to I(x, y, z) = I Pek ( ) exp ( x + y ) exp ( d 4xd ) exp ( ) When expressed in cylindricl coordintes (ρ, θ, z), Eq.(4) my be ritten s: I(x, y, z) = I Pek ( ) exp ( ρ d ) exp ( ) exp (4ρdcosθ ) (3) (4) (5)

5 To find the opticl poer P T trnsmitted through circulr perture, e integrte the irrdince of the displced bem over the entire perture of rdius : π P T = I Pek ( ) exp ( ρ P T = I Pek ( ) exp ( d ρ ) ρexp ( d ) exp ( ) exp (4ρdcosθ π ) dθ ρdρ (6) ) exp (4ρdcosθ ) dθ dρ P T = I Pek ( ) exp ( d ρ ) π ρexp ( ) I ( 4ρd ) dρ Where I (x) is the Modified Bessel Function of the first kind nd order hich cn be expnded to obtin n expression for P T s: (7) (8) P T = I Pek ( ) exp ( d ρ ) π ρexp ( ) ( k= P T = I Pek ( ) exp ( d ρ ) π ρexp ( ) P T = I Pek ( ) exp ( d ) π k= 4k d k k= 1 4 ( 4ρd k ) ) (k!) 4k d k dρ (9) 4k (k!) ρ k dρ (1) 4k (k!) ρ k+1 exp ( ρ ) dρ (11)

6 Solving the remining integrl results in: P T = I Pek ( ) exp ( d ) π k= [ k k ( ) 4k d k 4k (k!) k (Γ(k + 1) Γ (k + 1, ))] here Γ(s) is the stndrd Gmm Function nd Γ(s, f) is the upper incomplete Gmm Function. Simplifying nd rerrnging terms in Eq.(1) leds to: (1) P T = πi Pek ( ) exp ( d ) 4k d k 4k (k!) [ k k k k (Γ(k + 1) Γ (k + 1, ))] k= P T = πi Pek ( ) exp ( d ) (13) 4k d k 4k (k!) [ k k+ (Γ(k + 1) Γ (k + 1, ))] k= We lso kno from [5] the reltionship beteen the loer nd upper incomplete Gmm Functions nd the Gmm Function γ (k + 1, ) = Γ(k + 1) Γ (k + 1, ) Where γ(, x) is the loer incomplete Gmm Function. A simplified expression for the opticl poer of Gussin Bem trnsmitted through circulr perture is: (14) (15) P opt = π I Pek exp ( d ) k= k d k k (k!) (γ (k + 1, )) (16)

7 For the cse hen the lterl displcement of the Gussin bem is zero, the bem psses through the center of the perture, i.e. d =. We kno tht d k only for k =. For this cse the summtion in Eq.(15) simplifies nd e obtin: As e kno tht P opt = πi Pek ( ) (!) (γ (1, )) γ (1, ) = 1 exp ( ) Substitution of Eq.(18) in Eq.(19) leds to the ell-estblished expression for the trnsmitted opticl poer for non-displced Gussin Bem. (17) (18) P opt = π I Pek (1 exp ( )) The use of Eq.(16) to obtin this knon expression for trnsmitted poer through circulr perture for non-displced bem demonstrtes the vlidity of Eq.(16). The totl poer P totl of the Gussin bem is: (19) π P totl = I Pek ( ) exp ( ρ d ) exp ( ) exp (4ρdcosθ ) dθ ρdρ = π I Pek Substituting Eq.() in Eq.(16), e obtin n expression for poer trnsmittnce T of lterlly displced Gussin Bem through circulr perture s: () T = P Trnsmitted P totl = exp ( d ) k d k k (k!) γ (k + 1, ) k= (1)

8 3. Conclusion In this pper, the uthors hve presented mthemticl solution to fundmentl nd common problem in opticl systems. Most opticl systems rely on Gussin TEM mode propgtion nd it is commonly ssumed tht poer trnsmittnce through circulr perture is either very close to 1% for pertures rdii hich re much lrger thn the Gussin Bem ist. For smller circulr pertures, the centers of the bem irrdince distribution nd the circulr pertures re most ssumed to coincide to simplify clcultion of trnsmitted opticl poer through the perture. In this pper e hve presented n ccurte expression for poer trnsmission of lterlly shifted Gussin Bem through circulr perture. The expression explins vrition in poer trnsmission for ny lterl shifted Gussin Bem nd it simplifies to knon reltionship hen the displcement is set to zero. References 1. A. E. Siegmn, Lsers, pp. 666, University Science, E. Hecht, Optics, 4th edition, Addison Wesley,. 3. Arshd, Muhmmd Assd, Syed Azer Rez, nd Ahsn Muhmmd. "Dt trnsfer through bem steering using gile lensing." In SPIE Photonics Europe, pp Interntionl Society for Optics nd Photonics, Prrot SA Confidentil, Arctic 316 Arctic 316-AR85, Dvis, Philip J. "Gmm function nd relted functions." Hndbook of Mthemticl Functions (M (1965).