To the use of Sellmeier formula

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1 To the use of Sellmeier formula by Volkmar Brücker Seior Experte Service (SES) Bo ad HfT Leipzig, Germay Abstract Based o dispersio of pure silica we proposed a geeral Sellmeier formula for various dopats like GeO 2, P 2 O 5, B 2 O 3 ad F suitable for calculatio of dispersio i glass fibers. 1 Itroductio Already i 1871 Wolfgag Sellmeier [1] foud that the refractive idex depedece o wavelegth is very similar i may trasparet ad o-trasparet media i visible ad ear ifrared rage. It ca be described by the well-kow Sellmeier formula ( ) 1 M i1 ² Ai ² ² i Eq. 1 For M= 3 Eq. 1 was amed 3-term Sellmeier equatio. That meas measurig the refractive idex of the medium at least at 6 differet wavelegths we are able to calculate 6 Sellmeier costats A 1, A 2, A 3, λ 1, λ 2 ad λ 3 ad to approximate the dispersio curve. Today Sellmeier formula is widely used i optical sciece ad optical idustry to describe ad characterize the dispersio i glasses ad crystals. Ad may compaies which offer optical materials deliver their products together with the correspodig Sellmeier coefficiets, e.g. Schott [2]. As a example we ca use Sellmeier costats of the most popular glass BK7 [2] to describe the dispersio curve. I practical cases it is ofte ecessary to kow dispersio (λ) roughly that s why we tried to develop a more geeral Sellmeier formula which ca be used for may materials like a rule of thumb. 2 Glasses for fiber optics For fibers with core ad claddig(s) we eed special glasses, which have very low losses of about 0.2 db/km (at 1550 m) combied with well-kow refractive idices. To prepare special fibers ad/or calculate their trasmissio properties the refractive idexes of core ( core ) ad claddig ( clad ) as well as the correspodig dispersio properties should be flexible ad wellkow. I fiber optics, pure silica (SiO 2 ) with its specific depedece of refractive idex o wavelegth serves as the basic material. 2.1 Dopats of glasses For a log time [3] it is kow that the refractive idex ca be chaged by additives (dopats). To icrease the refractive idex, oe has to add, e.g., GeO 2, TiO 2 or P 2 O 5, to decrease it, e.g., B 2 O 3 or fluorie (F). Followig [3] we developed empiric formulas to describe this depedece o cocetratio d (i Mol%) of dopats: for GeO 2 Ge (d) = d Eq. 2 for TiO 2 Ti (d) = d for P 2 O 5 P (d) = d for B 2 O 3 B (d) = d d 2 Eq. 3 Eq. 4 Eq. 5

2 for F F (d) = d d 2 Eq. 6 Results are depicted i Fig. 1. As referece idex we used the refractive idex i SiO 2 at λ = 633 m TiO 2 GeO 2 P 2 O 5 B 2 O 3 F % Fig. 1 Chages of refractive idex i glass by additives To describe the depedece of refractive idex o wavelegths for differet dopats as fuctio of dopats cocetratio we used data from [4] ad obtaied the Sellmeier parameters give i Tab. 1. Tab. 1 Sellmeier coefficiets of glasses with differet dopats [4] Material Name i Fig. 2 A 1 λ 1 A 2 λ 2 A 3 λ 3 SiO 2 Si ,5% GeO ,5% SiO 2 Ge ,1% P 2 O ,9% SiO 2 P ,3% B 2 O ,7% SiO 2 B % F + 99% SiO 2 F Oe ca see from i Fig. 2 that the dispersio shapes for differet dopats are very similar ad oly shifted by a offset Δ. It should be oted, however, that other dopats, e.g. silica with 16.9% Na 2 O ad 32.5% B 2 O 3 lead to dispersio curves with other shapes. This ca be advatageously used for suitable core-claddig compositios to maage dispersio i the fiber. Ge P Si 1.44 F B Fig. 2 Dispersio i glasses with differet dopats

3 2.2 Variatio of cocetratio of GeO 2 dopats There exist a huge umber of papers which provide the Sellmeier coefficiets ad the correspodig dispersio for various udoped ad doped glasses. May authors studied the dispersio i silica (SiO 2 ) doped with GeO 2. I Tab. 2 we give the data of [5] = source 1, ad [6] = source 2. Tab. 2 Sellmeier coefficiets of silica for various GeO 2 cocetratios GeO 2 (mole%) source Sellmeier costat A 1 A 2 A 3 λ 1 λ 2 λ I Fig. 3 dispersio curves are give separately for two sources of Sellmeier coefficiets. Icreasig refractive idex correspods to icreasig cocetratio of dopats betwee % GeO 2 (Fig. 3a) ad % GeO 2 (Fig. 3b). As wavelegth rage we cosidered the rage of fiber optical widows ( µm). 19.3% 13.5% 0% 0% (a) (b) Fig. 3 Dispersio i silica for differet GeO 2 cocetratio from sources 2 (a) ad 1 (b) As oe ca see i Fig. 3 early all dispersio curves have the same shape. The oly differece is a offset i refractive idex Δ. For example we calculated the differece Δ betwee maximum

4 (19.3 % i Fig. 3a, 13.5% i Fig. 3b) ad miimum (0%) cocetratios of dopats GeO 2 (Fig. 4). Δ (a) (b) Fig. 4 Maximum differece Δ at GeO 2 cocetratios correspodig to Fig. 3a ad b Furthermore; at all wavelegths we foud a liear depedece of refractive idex o GeO 2 cocetratio calculated by Sellmeier formula. As a example, this depedece is illustrated for 1300 m i Fig. 5 separately for refractive idexes from source 1 ad source 2. A slight differece was oly foud i the slope ( /% for source 1 ad /% for source 2). source 1 source %GeO 2 Fig. 5 Depedece of refractive idex o cocetratio of dopats at λ = 1300 m Havig this behavior i mid we suggest a approximatio of these dispersio curves by a geeral formula. It starts from dispersio of pure silica, ad we have to add the offset i agreemet with dopats cocetratio. To this ed we have to select a referece wavelegth, where our choice is 1300 m. That meas our geeral formula is based o Sellmeier equatio of pure silica: ² Si ( ) 1Ai with A i ad i ² Eq. 7 i1 i ² The we have to add oly a offset depedig o the cocetratio of dopats (d Ge ) usig the slope of curve i Fig. 5 (we used the curve correspodig to source 2). For GeO 2 dopats we get the followig equatio: Ge (λ)= Si (λ) d Eq. 8

5 Calculatios usig this equatio are i very good agreemet with data described i the literature (see). It works perfectly i the wavelegth rage used i optical commuicatio techologies like DWDM ( m). Here the maximum error is less tha calculated (15%) source 2 for (15%) calculated (10%) Fig. 6 Dispersio curves for d Ge = 10% ad 15% GeO 2 calculated by usig Eq. 8 ad curve correspodig to source 2 for d Ge = 15% For practical use it is ofte useful to use derivatives of refractive idex, i.e. group idex d g ad group velocity dispersio (GVD) resultig i the material dispersio d d g d² parameter D Mat = = -. For pure silica it is depicted i Fig. 7. Note that d c c d² D Mat (λ) is exactly the same for all dopats ad cocetratios discussed here g g (15% GeO 2 ) g (10% GeO 2 ) D Mat (ps/m. km) 0 g (silica) Fig. 7 Group idex g i doped ad udoped silica ad dispersio parameter D Mat i pore silica

6 2.3 Variatio of cocetratios of other dopats Ufortuately, util ow here are ot available sufficiet experimetal data to make similar calculatios for other dopats (P 2 O 5, B 2 O 3 ad F). We ca use oly make use of dopats cocetratios give i Tab. 1. Agai we selected 1300 m as referece wavelegth. I this case the geeral formula is based agai o dispersio of silica ad we have to add a offset depedig o cocetratio of dopats. The slope ca be obtaied from Fig. 2 at 1300 m. Thus we get the followig formulas: for P 2 O 5 P (λ) = Si (λ) d P Eq. 9 for B 2 O 3 B (λ) = Si (λ) d B Eq. 10 for F F (λ) = Si (λ) d F Eq. 11 At λ = 633 m equatios 8 11 are i rather good agreemet with results of Fig. 1 correspodig empirical equatios 2 ad 4 6. Note that group idex ad group velocity dispersio behavior is i geeral the same as depicted i Fig Use of geeralized formulas Ofte there is aother task: To create ad produce special (maily sigle mode) fibers oe has to select material for core ad claddig(s) to achieve a certai umerical aperture 2 core 2 clad NA (applicatio-specific fibers). For example, we assume NA = 0.14 is required i stadard sigle-mode fibers SMF-28 of Corig Glass compay. Usig equatios 8 11 we are able to calculate for this fiber clad at give values of core ad NA, or core at give values of clad ad NA, respectively. For calculatio of ecessary cocetratio of dopats to realize a certai refractive idex 1 we developed the followig formulas (ote that 1 should be larger tha Si for GeO 2 ad P 2 O 5 ad less tha Si for B 2 O 3 ad F): for GeO 2 : for P 2 O 5 : for B 2 O 3 : for F: 1( ) Si( ) dge ( ) ( ) Si( ) dp ( ) d B Si( ) 1( ) ( ) Si( ) 1( ) df( ) Eq. 12 Eq. 13 Eq. 14 Eq. 15 Examples for calculatios usig equatios are give i Tab. 3. Here we used NA = 0.14 ad λ = 1550 m.

7 Tab. 3 Calculatios of core ad claddig refractive idexes Core Claddig core dopats cocetratio equatio cladd dopats coce- equ. (%) tratio (%) 14 GeO pure SiO 2 - Eq. 7 E q GeO GeO E q. 12 Eq P 2 O pure SiO 2 - Eq. 7 E q GeO P 2 O E q. 12 Eq pure SiO 2 - Eq B 2 O Eq pure SiO 2 - Eq F 1.5 Eq. 15 This algorithm ca be used for preparatio of applicatio-specific glass fibers. At the same time oe ca use the correspodig Sellmeier formulas to describe material, waveguide ad chromatic dispersio [7]. All calculatios have bee performed by the program MathCad. Correspodig programs may be dowloaded from the same olie-plus website at Vieweg+Teuber publisher. Refereces [1] Sellmeier W.: Zur Erklärug der aborme Farbefolge im Spektrum eiiger Substaze, Aale der Physik. 143, 1871, S [2] Schott data sheets: [3] Hultzsch, H. (Editor): Optische Telekommuikatiossysteme. Gelsekirche: Damm Verlag [4] Kerste, R. Th.: Eiführug i die Optische Nachrichtetechik. Berli Heidelberg New York: Spriger Verlag [5] Huyh T. L.: Dispersio i photoic systems, Techical report MECSE , Dept. of Electrical ad computer systems egieerig, Moash Uiversity, Clayto, Australia 2004 [6] Brücker V.: Elemets of Optical Networkig. Spriger Fachmedie Wiesbade GmbH: Vieweg+Teuber Verlag 2011 [7] Brücker V., Bylia M.S. et al.: Material, waveguide ad chromatic dispersio i differet sigle-mode fibers (i preparatio)

Ray Optics Theory and Mode Theory. Dr. Mohammad Faisal Dept. of EEE, BUET

Ray Optics Theory and Mode Theory. Dr. Mohammad Faisal Dept. of EEE, BUET Ray Optics Theory ad Mode Theory Dr. Mohammad Faisal Dept. of, BUT Optical Fiber WG For light to be trasmitted through fiber core, i.e., for total iteral reflectio i medium, > Ray Theory Trasmissio Ray