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1 Available olie at Archives of Physics Research,, (4):-7 ( ISSN : CODEN (USA): APRRC7 Dimesioless Itesity i the Semiclassical theory of Laser, Electric Field iside a Fabry-Perot Cavity ad Voltage i a riode Oscillator Circuit J. Saikia, H. Saikia, 3 R.K. Dubey* ad 4 G.D. Baruah Departmet of Physics, J.B. College, Jorhat, Assam Departmet of Physics, Majuli College, Majuli, Jorhat, Assam 3 Istitute of Educatio-Haldia,West Begal 4 Cetre for Laser ad Optical Sciece, New Uchamati, Doom Dooma ABSRAC he preset work reports a aalogy i three differet topics appearig i the semiclassical theory of laser, Fabry Perot Cavity ad triode oscillator circuit. We have show that the buildup of the parameter dimesioless itesity i the semiclassical theory ad the buildup of electric field iside a Fabry Perot cavity bear a strikig similarity. I this coectio Vader Pol s triode oscillator circuit is also discussed. INRODUCION he preset work is cocered with a comparative study betwee the buildup of dimesioless itesity i the semiclassical theory of laser ad the buildup of electric field due to multiple reflectios iside a Fabry Perot cavity. We have also brought uder our discussio the buildup of voltage i a triode oscillator circuit. It is worthwhile to ote that this type of work has ot bee reported earlier. A aalogy betwee spatial hole burig ad the itesity cotour of the beams multiple reflectios iside Fabry Perot cavity was established i a earlier work []. Recetly [] the subject of spatial hole burig, multiple reflectios iside a Fabry Perot cavity ad squeezed states of light have bee discussed ad i all the cases worthwhile aalogy amog the three pheomea has bee oted. We also ote here that i may ways some pheomea of physics appearig uder differet cotexts are quite aalogous. As for example the compariso betwee the laser ear threshold ad matter ear a phase trasitio was developed by DeGiorgio ad Scully (3) log back.. BUILD UP OF DIMENSIONLESS INENSIY he dimesioless itesity is a importat parameter obtaied i the semiclassical theory of laser (4, 5). he parameter is defied as I = Ε h γ γ a b

2 R.K. Dubey et al Arch. Phy. Res.,, (4):-7 a =..() β * Where = is the magitude of the electric-dipole matrix elemet. a is the term kow as liear et gai, that is υ a = L ( ω- υ ) F Q ( ) = + ( ) L ω- υ γ γ ω υ F = υ ( h γ ) N β = L ( ω υ ) F 3 F = ( 3 )( γ γ ) F 3 ab All these terms may be deduced from the semiclassical theory of laser (4). We are particularly iterested i the parameter I. he equatio of motio for the dimesioless itesity I is give by [4] as ( ) I = I a β I..() his itesity equatio of motio () ca be itegrated by the method of partial fractio with the solutio. [ ( a β I )] exp ( ) a I o o a t I ( t) = + β [ I ( a β I )] exp ( a t) o o..(3) Where I o I (o). µ is expected from this equatio that iitially this yields expoetial gai I ( t) = I exp( a t o ) for small I o ad it yields the steady state result I = a β. he time developmet is illustrated i Fig.. Fig. : Build up of dimesioless itesity for small value of ( ) I =.. It is worthwhile to ote that the buildup of itesity does ot take place from zero value.

3 R.K. Dubey et al Arch. Phy. Res.,, (4):-7 I this coectio we cosider the geometrical model based o which the semiclassical theory of laser is built. he geometrical model is used to describe the situatio iside a Fabry-Perot cavity ad self cosistecy priciple is ivolved. Fig. shows the geometrical model used by Lamb (4). Fig. :Geometrical Mode of Semiclassical heory. Electric filed E assumed i cavity iduces microscopic dipole momets (P) i the active medium accordig to the laws of quatum mechaics. hese momets are the summed to yield the macroscopic polarizatio of the medium P (r, t) which acts as a source of Maxwell s equatios. he coditio of self-cosistecy the requires that the assumed field E equal the reactio field E /. 3. FABRY-PERO RESONAING CAVIY We must emphasize that Fabry-Perot Cavity is a situatio of resoace of a electromagetic wave at optical frequecies which is o differet tha the resoace of ay other system, be it mechaical or electrical. here is always a iterchage of eergy i such a system betwee the potetial ad kietic forms i the case of a mechaical system, with attedat frictio losses, or betwee electric ad magetic eergy with resistive losses i a electromagetic problem. Quite ofte the pheomeo of resoace gets lost i the mathematics whe aalyzig a low frequecy system. However, fortuately a much simpler physical picture emerges whe we cosider systems where the wavelegth is much less tha the dimesios of the compoets. A example is the Fabry-Perot Cavity. I order to make the problem as familiar as possible we cosider the excitatio of the cavity as show i Fig.3, by a exteral source as a electric field i the form of a tuable laser or a variable frequecy oscillator. We cosider all waves, icidet o the cavity from the left, iside the cavity or trasmitted through it to the right to be uiform plae waves of limited spatial extet trasverse to the directio of propagatio. Our problem is to relate the fields, ruig wave itesities ad stored eergy o the iside of the cavity to those quatities that we ca measure o the outside. It may be oted here that we are dealig with a classical situatio aalogues to the semiclassical picture. Let us follow a wave as it bouces back ad forth betwee the mirrors. Cosider the iitial field at the plae just to the right of M, leveled by E. 3

4 R.K. Dubey et al Arch. Phy. Res.,, (4):-7 Fig.3 : Resoatig Cavity It propagates to M ad back to the startig plae ad experieces a amplitude chage of ρ, ρ ad a phase factor exp[ jk d] as it travels that roud trip ad thus geerators the field E +. his ew field experieces the same chage as E ad it i tur geerates E + ad so o. However there shall be a time whe the resultat filed will o loger experiece aother chage ad E + virtually remais costat. his is the coditio of self cosistecy i the geometrical model of the semiclassical theory of laser. Let us cosider agai the electric fields prior to the self cosistecy coditio. We ote that at every poit alog the path from the mirrors M to M, the fields E +, E +, E + ad so o are to be added to E to which we assig the referece phase of O. 3 his phasorial additio is show i Fig.4. Where, because there is a assumed laggig phase agle, we have assumed that the roud-trip phase shift (RPS) θ = kd is almost but ot quite a itegral multiple of π radius. he deficiecy is laveled by φ ad is related to kd by θ = kd = qπ φ. If the agle φ is sigificat, the total field propagatig to the right iside the cavity is the differece betwee the origi ad the spiral of the phasors quite similar to the straight lie distace betwee the begiig ad ed of a coiled rope. his distace is small, but if that rope is ucoiled, the distace is much larger. he distace will presumably deped o the total umber of coils ad the radius of the idividual coil. I a similar way the total field E will be may times the iitial value E, if ρ are close to uity ad φ =. he followig quatities are all, maximized by the simple equatio φ = ; the total field travellig to the right (ad thus to the left also); the magetic field associated with E, the itesities of the waves; the umber of photos boucig back ad forth ad the stored eergy. hese physical facts are characteristic of resoace which is give by Roud rip Phase Shift (RPS) = kd = kπ (4) 4

5 R.K. Dubey et al Arch. Phy. Res.,, (4):-7 Fig.4 : Phasor diagram illustrate the field up of electric field iside the resoatig cavity. his situatio is similar to the filed up of a parameter kow as dimesioless itesity i the semiclassical theory. Sice Where ω π k = =, we ca use it to fid the resoat wavelegth. C λ ω d kd = C π d = λ = q π λ = λ o or, qλ d =. his view of resoace states that there has to be a itegral umber of half wavelegths betwee the two mirrors. his also implies that the itegral q is a very large umber for optical frequecies ad reasoable size cavities. From Fig.3 oe ca estimate the total field E travellig to the right as follows : E = E + E + E +... ( ρ ρ ) = E + E ρ ρ e + E e + jk d jk d... α ( ) ( ) + jk d jk d jk d 3 = E = E + ρ ρ e + ρ ρ e + ρ ρ e +... E Or, E =.(5) j θ ρ ρ e he field returig from M is just ρ times the roud trip phase factor exp[ jkd ], multiplyig the wave goig to the right, E j = E = ρ e θ E + ρ e = E ρ ρ e j θ j θ..(6) 5

6 R.K. Dubey et al Arch. Phy. Res.,, (4):-7 From equatio (5) we ca also have a idea of buildup of electric field i the cavity, it may be oted that uder coditio of equilibrium of resoace the icidet field E should be equal to the resultat field E. I/I. otal Eergy Q Iitial Eergy with differet phase factors Fig.5 : he buildup of electric field. he buildup of electrical field as give by equatio (5) is show i Fig.5. As may be iferred from it the buildup exhibits a expoetial growth with a saturatio at a latter stage. hus the dimesioless itesity i the semiclassical theory of laser correspods to the electric field i the Fabry-Perot cavity. 4. VANDER POLE S RIODE OSCILLAOR Vader Pole (6) obtaied a equatio of motio i his treatmet of the triode oscillator as d ( av β υ ) && / 3 + ω υ =.. (7) V dt / Here a is the liear et gai (i.e. the gai i excess of losses), β is the saturatio co-efficiet ad ω is the resoace frequecy i the absece of dissipatio or gai. Equatio (7) may be solved approximately by a importat techique called the method of slowly varyig amplitude, writig υ ( t) = V ( t) exp ( iωt ) + C. C... (8) Where V(t) is the amplitude. he slowly varyig equatio of motio is obtaied as V t V t a V ( ) = ( )( β ) &.. (9) 6

7 R.K. Dubey et al Arch. Phy. Res.,, (4):-7 For sufficietly large V(t), saturatio sets i ad the steady state coditio V& ( t ) = Occurs for the value V = a.. () β Aalytical solutio of equatio (9) is discussed by Lamb i the semiclassical laser theory ad the solutio is give by equatio (3). It may be oted that equatio (7) ad hece (9) does ot build up from the zero amplitude. I practice some fluctuatios get thigs goig. RESULS AND DISCUSSION From what has bee described above it is worthwhile to make a coclusio ad idicate the saliet features. We have briefly describe the geometrical model based o which the semiclassical theory of laser is built. he mai characteristic is the self cosistecy ature of the icidet field E ad the resultat field E /. his also describes the threshold coditio of oscillatio for laser actio idicatig how loss compesates gai. A parameter kow as dimesioless itesity is defied ad it is show how build up of this itesity takes place iside the cavity. he etire situatio is co-related with aother picture of resoatig cavity where it is show how i a multiple reflectio the electric filed udergoes chages. A phaser diagram has also bee used to describe the situatio. Uder coditio of equilibrium this must idicate the essetial features of the geometrical model, because the self cosistecy ature is othig but a coditio of resoace. he Vader Pol s oscillator circuit also describes the build up of voltage like the build up of dimesioless itesity i the semiclassical theory. I this case we have ot cosider a fully quatum mechaical versio of laser where it is show how build up photo desity takes place from zero value. REFERENCES [] R.M. Borah ad G.D. Baruah, Pramaa, Joural of Physics, 54, 69 () [] J. Saikia, R.K. Dubey ad G.D. Baruah, Archives of Physics Research (), 64 7 () [3] V. De Giorgio ad M.O. Scully, Phys. Rev. A, 7 (97) [4] W.E. Lamb Jr. Phys. Rev. 34, A49 (964) [5] M. Sarget III, M.O. Scully ad W.E. Lamb Jr., Laser Physics, Addiso Weslwy Publishig Compay, Readig, Massachusetts, (974) [6] Vader pol., B. 96, Radio Rev., , Veder pol.b., Phil Mag.3, 65. [7] M.O. Scully ad W.E. Lamb Jr., 967, Phyc. Rev. 59, 8. 7