Theoretical Study of Electromagnetic Wave Propagation: Gaussian Bean Method

Transcription

1 Applid Mathmatics, 3, 4, Publishd Onlin Octob 3 ( Thotical Study of Elctomagntic Wav Popagation: Gaussian Ban Mthod E. I. Ugwu, J. E. Ekp, E. Nnaji, E. H. Uguu Dpatmnt of Industial Physics, Ebonyi Stat Univsity, Abakaliki, Nigia ugwui@hahoo.com Rcivd Mach, 3; visd Apil, 3; accptd Apil 9, 3 Copyight 3 E. I. Ugwu t al. This is an opn accss aticl distibutd und th Cativ Commons Attibution Licns, which pmits unstictd us, distibution, and poduction in any mdium, povidd th oiginal wok is poply citd. ABSTRACT In this wok, w psnt th study of lctomagntic wav popagation though a mdium with a vaiabl dilctic E function using th concpt of Gaussian Bam. Fist of all, w stat with wav quation E with t which w obtain th solution in tms of th lctic fild and intnsity distibutions appoimat to Gaussian Function, I, y. With this, w analy th dpndncy of on Gaussian bam distibution spad, th distant fom th ais at which th intnsity of th bam distibution bgins to fall at a givn stimat of its pak valu. Th influnc of th optimum bam waist w o and th bam spad on th intnsity distibution will also b analyd. Kywods: Elctomagntic Wav; Wav Equation; Dilctic Mdium; Distibution; Gaussian Function; Intnsity; Elctic Fild; Bam Waist; Wav Popagation. Intoduction Along with th apid volution of fib optics, intgatd optics and application of las in both mdicin and tchnology, th has bn a gowing intst in th study of Gaussian bam popagation. This is basd on th fact that Gaussian bam has to do with focusing and modification of shap of popagating lctomagntic wav o las bam. Fom th ali finding on th sach on las bam, it has bn found that las bam popagation can b appoimatd by an idal Gaussian bam intnsity pofil [,]. Undstanding of th basic poptis of Gaussian bam has bn spcifically found to b vy vital. I slct th bst optics fo pactical application [3]. Squl to this, lots of scintists had wokd on Gaussian bam applications in lctomagntic wav popagation and in optics. Fo instanc, a wok has bn caid out on nonspcula phnomna fo bam flction at monolay and multilayd dilctic intfac spctivly fom wh it has vald that und vaious conditions nonspcula bam phnomna is mo aliabl [4-6]. Tami on his own psntd a unifid and simplifid analysis of th latal and longitudinal displacmnt with angula dflction on flction of a Gaussian bam at a dilctic intfac with two o mo lay [7,8]. Howv in this pap, w intnd to study lctomagntic wav popagation using th concpt of Gaussian bam stating fom gnal wav quation with which obtain th Gaussian function in which wavs opating on th fundamntal tansvs mod is appoimatd to Gaussian pofil. Th distibution pofil is analyd with intnt to obsv th influnc of th bam waist on th pofil.. Thotical Famwok In this cas w stat with wav quation in tms of lctic fild givn as E E () t wh is th dilctic constant as a function wav solution of th fom E E,, y, pi tk () Which if w consid mdium with non-unifom - Copyight 3 SciRs.

2 E. I. UGWU ET AL. 467 factiv ind, w hav E wh th popagation constant in th mdium is givn by k (4) (3) With this, Equation () bcoms E k E (5) In gnal, if th mdium is absobing o hibit gain, th its dilctic constant is considd to hav al and imaginay pat, but in a situation wh th mdium conducts with conductivity. thn th compl popagation vcto is intoduc which obys. k i [9]. (6) Howv, a simpl solution of this typ is inadquat to dscib th fild distibutions of tansvs mod. As a sult, w sk solution to th plan wav quation of th fom. E E, y, pik (7) Popagating in th -diction and localid th -ais. Thus th ida is to obtain a solution of th wav quation that givs phas font that can b appoimatd ov a naow gion. Thus substituting Equation (7) into Equation (4) w obtain. E E E E ik (8) y Sinc w a looking fo a paaial bam-lik solution, thn E vais slowly with and Equation (8) bcoms E E E ik y With a solution givn as k E, y, pip w (9) () y y fom th ais of popagation wh a bam paamts Equation () givs th fundamntal Gaussian bam of tim-indpndnt wav quation. wh is th squa distant of th points, p q 3. Intnsity of Gaussian Bam Th intnsity of I, y of Gaussian bam is givn by I EE y, k p p ip ip q q k () wh p and q a th compl conjugat of p and q spctivly hnc ik EE p ip p p () q q Considing two al bam paamts that R and lating to q by dpnd on i, (3) q R w p is that dscibs th aial distanc fom th Gaussian bam waist. W can pss Equation () as EE I p y ik i i (4) w w p (5) w Signifying th dpndncy of Gaussian bam intnsity to whil w signifis th distanc fom th ais at wh th intnsity of th bam falls to of it its pak valu on th ais p total pow of th bam with ths paamts Equation () is now wittn as k i E p i k p R (6) p is valuat to giv q q (7) wh q is a constant of intgation th psss th valu of th bam paamt at th plan fo dp i which whn w consid, w obtain d q q In o p i q iinq (8) wh iin q dfins a constant of intgation substituting Equation (8) into (6) w hav k i E p ik iin q R (9) Th facto p i p is a constant phas facto. Thus k i E p ik iin q R 4. Diffaction Effct () Th limitations of Gaussian bam basd on th fact that vn if th wav fonts w mad flat at som plan, it Copyight 3 SciRs.

3 468 E. I. UGWU ET AL. quickly acquis cuvatu and bgin to spad in accodanc with and R w w () w () w wh is th distanc popagatd fom th plan wh th wav-font is flat wavlngth of light, w is th adius of th iadianc contou at th plan wh wav-font is flat, w is th adius of th contou aft th wav has popagatd a distanc. Considing th optimum stating bam adius fo a distanc, w hav b ducd to, w Equation () can w 3 w (3) With th iadianc distanc distibution of th Gaussian bam dscibd as p I I p p (4) w wh w w and p is th total pow in th bam which is th sam at all coss sctions of th bams w also obtain that I 3 In (5) I w I In 3 y I (6) at optimism stating bam adius fo a givn distanc,. 5. Discussion p I w (7) w Fom th displayd sults in Figu, th distibution pofil whn y =, = and w =, th distibution bam width angd within 5 to 5, In Figu, fo y = and = 3 whn w =, th width of th bam angd fom 6.5 to 6.5. In Figus 3 and 4 th distibution pofils displayd is found to b dpatd fom nomal distibution shap as in th oth figus whn w is incasd to 5 fo Figu. Gaussian distibution pofil whn y =, w = Figu. Gaussian distibution pofil whn y =, w = 4. g( ) 5 5 Figu 3. Gaussian distibution pofil whn y =, w = 5. y = whn w =, th dpatu bcam mo ponouncd pattn as shown in Figu 5. Howv, th distibution pofil of pcntag iadianc as function Copyight 3 SciRs.

4 E. I. UGWU ET AL. 469 f ( ) w is Figu 8 ov a distant and a paticula wavlngth. 6. Conclusions Gaussian intnsity pofils as obsvd fom this study sm to manifst on fatu that is basd on th fact that bam waist, w plays a ol in dtmining th distibution pofil. This plains th fact that th fa-fild divgnc 5 5 Figu 4. Gaussian distibution pofil whn y =, w = 5. f ( ) Figu 6. Pcntag iadianc. 5 5 Figu 5. Gaussian distibution pofil whn λ =.5 m, y =, w =. contou adius as displayd in Figu 6 indicat th fact that as th contou dcass, th bam distibution tapps showing that th is a point known as spot si wh th intnsity of th distibution is falln to. This indicats that Gaussian distibution function dpnds on as shown in Equation (6) that dpicts chaactistic of nomal distibution function. Howv, th shap of th distibution dpnds gn- Figu 7. Gaussian distibution intnsity as function of pofil width, w. ally on diamt at which th intnsity has falln of as in Figu 7 whn = y coupld with th optimum bam waist that is givn as wo optimum, aial distanc and Raligh ang which is givn in Equa- wo tion () with R [9] Ths valus povid th bst combination fo minimum stating bam diamt fo good minimum spad accoding to [] whos atio Figu 8. Gaussian bam width w() as a function of th aial distanc. Copyight 3 SciRs.

5 47 E. I. UGWU ET AL. must b masud at a distanc gat than th half width of intnsity distibution whil th na o mid-fild divgnc valus a obtaind by masuing a distanc lss than th half width of th intnsity distibution in od to obtain a good nomal distibution pofil. Thfo in od to achiv optimum bam spad, and collimation ov a distanc, th optimum stating bam adium/waist must b dtmind. Though on should not that this dpnds stictly on th typ of bam that is whth th bam is cohnt o not, th gnal pssion fo optimum stating bam adius fo any givn distanc, is w optimum. This study also vals that th concpt of Gaussian bam is aliabl whn it is focusd on a small spot as it spads out apidly as it popagat away fom th spot Fom Figus 3 to 6, it is claly shown that at ctain valu of th bam waist, th distibution pofil changs. Thfo fo a bam to b wll collimatd, it must hav a lag diamt. This lation is as a sult of diffaction. Thus in od to achiv nomal Gaussian bam distibution, th poduct of th width and divgnc of th bam pofil must b as small as possibl to b achivd whn paaial appoimation is considd. Thus this implis that th ida of Gaussian bam modl is aliabl and valid only fo bam whos width is lag than that of quation. REFERENCES [] A. Sigman, Lass Snsitiation, C.A Univsity Scincs Books, 986. [] S. A. Slf, Focusing of Sphical Gaussian Bam, Applid Optics, Vol., No. 5, 983, pp [3] H. Sun, Thin Lns Equation fo a Ral Las Bam with Wak Lns Aptu Tuncation, Optical Engining, Vol. 37, No., 998, pp [4] J. W. Ra, H. L. Btoni and L. B. Flsn, Rflction and Tansmission of Bam at Dilctic Intfac, SI Jounal on Applid Mathmatics, Vol. 4, No. 3, 973, pp [5] M. McGuiik and C. K. Caniglia, An Angula Spctum Rpsntation Appoach to th Goos-Hänchn Shift, Jounal of th Optical Socity of Amica, Vol. 67, No., 977, pp [6] R. P. Ris and R. Simon, Rflction of a Gaussian Bam fom a Dilctic Slab, Jounal of th Optical Socity of Amica A, Vol., No., 985, pp [7] T. Tami, Nonspcula Phnomna in Bam Fild Rflctd by Multilayd Mdia, Jounal of th Optical Socity of Amica A, Vol. 3, No. 4, 986, pp [8] S. Z. Zhang and C. C. Fan, Nonspcila Phnomna on Gaussian Bam Rflction at Dilctic Intfac, Jounal of th Optical Socity of Amica A, Vol. 5, No. 9, 988, pp [9] F. Pampaloni and J. Endlin, Gaussian, Hmit- Gaussian, and Lagu Gaussian Bam: A Pim, 4, 9p. [] J. Ralton, Gaussian Bams and th Popagation of Singulaity, Studis in Patial Diffntial Equations, MAA Studis in Mathmatics, Vol. 3, 98, pp Copyight 3 SciRs.