F. J. Duarte (005 www.opticsjoural.com/equatiosituablelaseroptics.pdf Equatios i tuable laser optics: brief itroductio F. J. Duarte Iterferometric Optics, Rochester, ew York, USA ECE, Uiversity of ew Mexico, ew Mexico, USA A succict itroductio to the various iterferece, diffractio, dispersio, ad liewidth equatios ecessary to predict the coherece properties of arrow-liewidth tuable lasers is give. The quatum mechaical origi of the cavity liewidth equatio, diffractio, refractio (Sell s law, ad reflectio, is also refereced.. Itroductio The equatios icluded here form the theory that is applied i the desig of practical high-power tuable laser oscillators regardless of the type of gai media. I other words, the theory applies for lasers usig tuable gai media i the gas, the liquid, or the solidstate. These equatios ca be used to either predict the value of real measurable variables i laser physics or, alteratively, they ca be used to explai the value of measured parameters such as laser beam divergece (Δθ ad laser liewidth (Δν. Both these parameters are essetial to determie ad characterize the coherece of a laser. The itroductio to this subject, give here, is backwards to the chroological developmet of the physics. But that should ot be surprisig sice, after all, Dirac s otatio is backwards. The readers should be aware that ot all the terms might be explaied here ad that cosultatio with the origial refereces is ecessary. All these equatios are explaied i detail, with the ecessary geometry ad schematics, i Tuable Laser Optics.. Geeralized Dirac iterferece equatios: iterferometric quatum origi of the liewidth equatio ad the ucertaity priciple The geeralized probability distributio for propagatio from a source s to a iterferometric plae x, is give by,
F. J. Duarte (005 www.opticsjoural.com/equatiosituablelaseroptics.pdf z= y= x= q= p= r= i( Ωqpr Ω zyx x s = Ψ( r Ψ( r e ( zyx qpr This equatio applies either to the propagatio of a sigle photo or to the propagatio of a large umber of idistiguishable photos, as i the case of laser radiatio. 3, I oe dimesio this equatio reduces to,5 x s = Ψ( r + Ψ j ( rj Ψ( rm cos( Ω m Ω j ( j= j= m= j+ I this equatio it is the cosie term that cotais all the iformatio about the geometry ad the wavelegth of the emissio. -6 This iterferece equatio has bee show to predict measured iterferometric distributios either i the ear or the far field. -5 Also, it ca be used to predict diffractio profiles due to trasmissio via sigle-slits. This is,,, 5 doe by dividig the sigle slit ito a large umber of imagiary sub slits. Further uses of this equatio iclude the derivatio of the liewidth cavity equatio via quatum priciples, 6 Δ Δθ ( θ / (3 that ca also be expressed as 7 Δ Δθ ( θ ( This equatio cotais all the essetial iformatio ecessary to desig arrow-liewidth tuable lasers. This equatio tells us that the liewidth of a pulsed laser is directly proportioal to its beam divergece (Δθ ad iversely proportioal to the itracavity dispersio ( θ. For the case of a laser cavity icludig a multiple-prism gratig assembly the multiple retur-pass liewidth is give by, 8 ( MR Θ + R Φ Δ = Δθ (5 R G
F. J. Duarte (005 www.opticsjoural.com/equatiosituablelaseroptics.pdf 3 where M is the itracavity beam expasio, R is the total umber of retur passes to the oset of laser emissio, ΘG is the dispersio of the gratig, Φ is the double-pass prismatic dispersio, ad 9 ( + ( L / B ( A L / B / Δθ R = ( / πw R R + R R R (6 is the geeralized expressio for the beam divergece which is a fuctio of the Rayleigh legth ad propagatio matrix terms. Equatio ( also explais the physics behid femtosecod, or ultrafast, lasers. I that case the itracavity dispersio is reduced to a miimum thus allowig for broadbad emissio ad hece, via the ucertaity priciple, to ultrashort pulse emissio. At this stage it is appropriate to idicate that the iterferometric equatio (Eq. ( ca be used to yield a approximate derivatio of eiseberg s ucertaity priciple 0- Δ xδp h (7 which, i tur, ca be used to derive a expressio for the diffractio limit of beam, 3 divergece Δ θ / πw (8 where w is kow as the beam waist. This is the miimum expressio for beam divergece uder ideal coditios. I practical lasers, as depicted i Eq. (6, this expressio is multiplied by the square root of a series of terms derived from the geometry of the resoator Uder ideal circumstaces that term reduces to ~. Thus we have described, very succictly, how geeralized iterferometic equatios derived usig the Dirac otatio ca be used to yield all the fudametal cocepts ecessary to desig arrow-liewidth tuable laser oscillators ad femtosecod, or ultrafast, lasers. Further, this approach has bee used to provide a uified quatum descriptio of optics i the followig order: iterferece, diffractio, refractio (Sell s, 3, 5 law, ad reflectio. ext we cosider the dispersio term i the liewidth equatio.
F. J. Duarte (005 www.opticsjoural.com/equatiosituablelaseroptics.pdf 3. Geeralized multiple-prism gratig dispersio equatios The subject of multiple-prism dispersio was first discussed, i a qualitative maer, by ewto i his prophetic book Opticks. Subsequetly, Brewster described the use of prism pairs. 5 owever, a geeralized mathematical descriptio of multiple-prism beam expaders ad their dispersio was oly made available followig the evet of the tuable laser. 6 Detailed reviews o this subject are give elsewhere. 7-0 Also, the 5, quatum origi of refractio has bee described elsewhere. Briefly, usig the law refractio (Sell s law as a starig poit, for a multiple-prism array, the cumulative, 6-0 sigle-pass dispersio at the mth prism is give by ( ±, m =, m m + ( k, mk, m, m m,( m (9 this is a recursive fuctio that depeds o the dispersio of the previous prism (m ad where the k,m term, for istace, refers to the beam expasio udergoe by the light beam followig icidece o the first surface of the mth prism. Also, m is the refractive idex of the mth prism ad, m,, m are additioal geometrical terms defied i the refereces. A explicit equatio for the double-pass itracavity dispersio, icludig terms describig the overall beam expasio M ad M, is give by 9 Φ r r r M ( ±, m k j k,, j m= j = m j = m = M m r m m ( ± m k j k,,, j m= j= j= (0 + m for idetical prisms deployed at Brewster s agle of icidece this equatio reduces to the succict expressio 9 Φ r m= m ( m m = ( ± (
F. J. Duarte (005 www.opticsjoural.com/equatiosituablelaseroptics.pdf 5 ow, goig back to the issue of pulse compressio: a proper discussio of this pheomeo requires a mathematical descriptio of both the first order dispersio, ad the secod order dispersio, m =, m, m (, m m + ( k, m k, m { χ k ± + ( ± (, m, m, m, m m,( m, m ( m ( χ, m ψ, m, mk, m ψ, m χ, mk, m, mm } + ( k, m ( ψ, m + ψ, m ( Further, it should be metioed that multiple-prism arrays were first described i matrix form i 989. 3 It should also be metioed that the geeralized beam expasio coefficiet ad dispersio, ca be itegrated as compoets of propagatio, matrices as described elsewhere. I order to iclude the case of egative refractio i the geeralized dispersio descriptio Eq. (9 takes the more geeral form of ( ( ±, m = ±, m m ± ( k, mk, m, m m,( m (3 where the sigs i parethesis refer to the geometrical cofiguratio whilst the simple ± refers to either positive (+ or egative ( refractio. Albeit substatial progress had bee made towards the mathematical/theoretical represetatio of the geeralized multiple-prism dispersio 6, complete access to higher phase derivatives has oly recetly bee grated For istace, the 5 th derivative of the geeralized multiple-prism refractio, or the th derivative of the geeralized multipleprism dispersio, is elegatly give by 5
F. J. Duarte (005 www.opticsjoural.com/equatiosituablelaseroptics.pdf 6 5, m =, m + ( M (, m ±,( m 3 + ( M + 6( M ( (, m, m ± ± 3,( m,( m ( + ( M ( 3, m ±,( m + ( M (, m ± 5,( m I this most curret extesio to the geeralized multiple-prism dispersio theory a clear ad elegat mathematical framework is provided to express, at will, ay higher derivatives up to the th order. 5 I referece to Eq. (, for istace, the kee observer ca recogize, from the secod to the fifth term, the elemets of ascal s triagle for the power of. Refereces. F. J. Duarte, Tuable Laser Optics (Elsevier Academic, ew York, 003.. F. J. Duarte, Iterferometric imagig, i Tuable Laser Applicatios, F. J. Duarte (Ed. (Marcel-Dekker, ew York, 995 pp. 53-78. 3. F. J. Duarte, Commet o "Reflectio, refractio, ad multislit iterferece," Eur. J. hys. 5, L57-L58 (00.. F. J. Duarte, O a geeralized iterferece equatio ad iterferometric measuremets, Opt. Commu. 03, 8- (993. 5. F. J. Duarte, Iterferece, diffractio, ad refractio, via Dirac's otatio, Am. J. hys. 65, 637-60 (997.
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