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 Theory Trasmissio (a) Refractio (b) Limitig case of refractio (c) Total iteral reflectio φ Agle of icidece φ Agle of refractio For case (a) For case (b) For case (c) si si Acceptace agle si c,, ow is agle of reflectio c Total iteral reflectio occurs whe icidece agle > critical value (φ c ) φ i = 90 0 - φ c oly this light may be cosidered to be propagated dow the fiber 3
Acceptace Agle Oly the meridioal rays over the rage 0 θ θ a will propagate withi the fiber If φ i > θ a (like ray B), light will be lost by radiatio So the ray to be trasmitted by total iteral reflectio withi the fiber core they must be icidet o fiber core withi a acceptace coe defied by coical half agle θa θa is the maximum agle to the core axis at which light may eter to the fiber i Order to be propagated θa is called the acceptace agle
Numerical Aperture NA gives the relatio betwee acceptace agle ad refractive idices of the three media ivolved: core, claddig ad air NA idicates the light collectig capacity of fiber At air-core iterface, usig Sell s law, si si 0
NA Calculatio From right-agled triagle ABC, 0 si si cos si whe, (limitig case) si si a 0 a si c c 0 sia NA si 0 a si a c
Relative Refractive Idex Differece Meriodioal Ray NA for It passes through the axis of fiber core
Skew Ray These rays follow a helical path through the fiber
Step-Idex Fibers r r a (core) r a (claddig) (r) profile makes a step chage at the core-claddig iterface
Multimode fiber: Supports may modes, core diameter ~ 50 μm Sigle Mode Fiber: Supports oly oe mode, core diameter ~ 0 μm
Graded Idex (GRIN) Fibers, r a (core), r a (claddig) r a r α is profile parameter α = step idex profile α = parabolic profile (best choice) α = triagular profile
Graded Idex (GRIN) Fibers Costat idex i claddig, but a decreasig idex with radial distace from a maximum value of
Multimode GRIN fibers exhibit less itermodal dispersio tha multimode SI fibers: may modes are excited but their differet group velocities ted to be ormalized by idex gradig. Rays travelig close to fiber axis have shorter paths but they travel with lower velocities as this is a regio of higher. Rays which travel ito the outer regio of core follow loger paths with higher velocities as is lower here. That s why, MM GI fibers with parabolic -profile i core have greater BW tha that of MM SI.
Compariso Sigle Mode SI fiber Multimode SI fiber MM GRIN fiber No itermodal dispersio Cosiderable itermodal dispersio Much less itermodal dispersio tha MM SI Huge BW Less BW Relatively greater tha MM SI LASR has to be used (spatially coheret light sources) LD ca be used (spatially icoheret light sources) Larger NA asier couplig as larger core diameter LD ca be used NA is smaller compared to MM SI with same Δ asier couplig as larger core diameter Problem: A typical relative refractive idex differece for a optical fiber desiged for log distace trasmissio is %. stimate the NA ad acceptace agle i air for the fiber whe core idex is.46. Further, calculate critical agle at the core-claddig iterface withi the fiber.
Wave Propagatio Propagatio of light i SI fibers usig Maxwell s equatio: For a o-coductig medium without free charges B t D H t D 0 B 0 No free charge No free pole D B Where ad H are electric ad magetic field vectors, respectively, D ad B are the correspodig flux desities. ε ad μ are permittivity ad permeability of the medium, respectively. By takig curl of first oe, B H t t t Usig the idetity t H
H Fourier Trasform of t H t Wave equatio i frequecy domai Where,, it r r t e dt r r0 0 c k 0 c 0 0 0 0 r Wave equatio 0 r k refractive idex of medium k 0 free space wave umber c 0 0 0
Fiber Modes A optical fiber mode refers to a specific solutio of wave equatio that satisfies the appropriate boudary coditios ad has properties that its spatial distributio does ot chage with time. Fiber modes: Guided modes Leaky modes Radiatio modes Wave equatio i cylidrical coordiates: z z z z k 0 z r r r r z 0 This equatio is for axial compoet z, similar equatios ca be writte for other five compoets of ad H. z ad H z are chose as idepedet compoets, r, ϕ, H r, ad H ϕ are depedet compoets For step-idex fiber with core radius a, r a (core) r a (claddig)
Fiber Modes The above equatio ca be solved by usig method of separatio of variables. The geeral solutio,, im i z z r z F r e e β is propagatio costat ad m is a iteger; d F df m k 0 F dr r dr r 0 This is well-kow differetial equatio satisfied by Bessel fuctios. Its geeral solutio iside core ad claddig regios: F r m AJ m pr A Y pr ; r a i core regio CKm qr C Im qr ; r a i claddig regio Where A, A, C,C are the costats ad J m, Y m, K m ad I m are the differet kids of Bessel fuctios ad
Fiber Modes p q k i core regio 0 k 0 i claddig regio Usig boudary coditios: Optical field should be fiite at r = 0 ad decays to 0 as r. Y m (pr) has a sigularity at r = 0. so A = 0 for F(0) to be fiite. Similarly, F(r) 0 at r oly if C = 0. Hece the geeral solutio: im iz AJ m pr e e ; r a CKm qr e e ; r a z im i z F r AJ m pr ; r a CKm qr ; r a Similarly, H im iz BJ m pr e e ; r a DKm qr e e ; r a z im i z
Other four (depedet) compoets by usig Maxwell s equatio (i core regio): z z r z z z z r z z H i p r r H i p r r H i H p r r H i H p r r These 6 equatios express the field i core ad claddig regios i terms of 4 costats A,B,C, ad D. These costats ca be determied cosiderig the boudary coditio that the field is cotiuous at r = a (at core-claddig iterface). Usig the requiremet of cotiuity of z,, H z, ϕ ad H ϕ at r=a We obtai a set of four homogeeous equatios satisfied by A,B,C,D.
These four equatios have a otrivial (ozero) solutio oly if the determiat of the coefficiet matrix vaishes. This coditio leads to the followig eigevalue equatio (it has may solutios): J m pa K m qa J m pa K m qa pjm pa qkm qa pjm pa qkm qa Where, 0 m a p q p q mk a p q p q k 0 Prime idicates Differetiatio with respect to the argumet For a give value of k 0, a,,, the eigevalue equatio ca be solved umerically to determie β. This eigevalue equatio may have multiple solutios for each iteger value of m.