- 1. 1 Fiber Optics Ray Theory SOLO HERMELIN Updated: 17.06.06 http://www.solohermelin.com
- 2. 2 SOLO Optical Fibre – Ray Theory http://www.datacottage.com/nch/fibre.htm
- 3. 3 A step-index cylindrical fiber has a central core of index ncore surrounded by cladding of index ncladding where ncladding < ncore. SOLO Optical Fiber – Ray Theory Cladding Core axisθ 0θ i θ Core axis Cladding Skew ray in core of fiber Meridional ray in core with two reflexions When a ray of light enters such a fiber at an angle θ0 is refracted at an angle θ, and then reflected back at the boundary between core and cladding, if the angle of incidence θi is greater than the critical angle θc. Two distinct rays can travel inside the fiber in this way: • meridional rays remain in a plan that contains fiber axis • skew rays travel in a non-planar zig-zag path and never cross the fiber axis
- 4. 4 For the meridional ray SOLO Optical Fiber – Ray Theory Cladding Core axisθ 0θ iθ Meridional ray in core with two reflexions Snell’s Law at the fiber enter If the ray is refracted from the core to the cladding than according to Snell’s Law: 222 0 sin1cossinsin claddingcoreicoreicorecore nnnnn −<−=== θθθθ r core cladding i n n θθ sinsin = If there is no tunneling from core to cladding.1sin:sin ≤=> c core cladding i n n θθ Since we have 90=+ i θθ θθ sinsin 0 1 coreair nn = Therefore total internal reflection will occur if: 2 22 0 1sin −=−< core cladding corecladdingcore n n nnnθ
- 5. 5 We consider only two types of optical fibers: SOLO Optical Fiber – Ray Theory Skew ray in step-index core fiber Meridional ray in step-index core fiber Core axis Cladding Core axis Cladding zθ φθ φ1 r1z1.constnn corecladding =< Meridional ray in a grated-index core Core axis Cladding Skew ray in a grated-index core of fiber ( )rnncore = Core axis Cladding zθ φθ r r1 φ1 • step-index core fiber where the index of refraction in core is constant and changes by a step in the cladding such that corecladding nn < • graded-index core fiber where the index of refraction in core changes as function of radius r such that ( )rnncore =
- 6. 6 For a graded-index core fiber ncore = n ( r ) let develop the ray equation: SOLO Optical Fiber – Ray Theory ( ) ( ) ( ) rrn rd d rn sd rd rn sd d 1 ray =∇= zzrrr 11ray += where: rayr -ray vector rayrdsd = Assuming a cylindrical core fiber we will use cylindrical coordinates zzddrrrdrd 111ray ++= φφ Graded-index Fiber sz sd zd sd d rr sd rd sd rd 1:111 ray =++= φ φ = −= = 01 11 11 zd rdd drd φφ φφ 011111 =−== z sd d r sd d sd d sd d r sd d φ φφ φ = +−= += zz yx yxr 11 1cos1sin1 1sin1cos1 φφφ φφ to describe the ray vector: ( ) ( ) ( ) ( )22222/1 zddrrdrdrdsd rayray ++=⋅= φ ray propagation direction See S. Hermelin, “Foundation of Geometrical Optics”
- 7. 7 SOLO Optical Fiber – Ray Theory Skew ray in core of fiber z θ φθ φ1 r1 z1 ρ Q P zrrr zzz 1cos1cossin1sinsin1 ray θθθθθ φφ ++= ρ φθ Core Q' axis Core axis Cladding zθ φθ r r1 φ1 ray1r ( )rnncore = ( ) ( ) ( ) rrn rd d rn sd rd rn sd d 1 ray =∇= Graded-index Fiber (continue – 1( z sd zd sd d rr sd rd sd rd 111 ray ++= φ φ ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 ray 1 1 1 1 1 1 sd zd sd zd rnz sd zd rn sd d sd d sd d rrn sd d rrn sd d sd rd sd rd rnr sd rd rn sd d sd rd rn sd d + + + + + = φφ φ φ ( ) ( ) ( ) ( ) ( ) z sd zd rn sd d r sd d rnr sd d rnr sd d sd d sd rd rnr sd rd rn sd d 11111 2 + − ++ = φ φ φ φ φ ( ) ( ) ( ) ( ) ( ) ( ) ( ) r rd rnd z sd zd rn sd d sd d sd rd rn sd d rn sd d r sd d rnr sd rd rn sd d sd rd rn sd d 11121 2 ray = + + + − = φ φφφ 011111 =−== z sd d r sd d sd d sd d r sd d φ φφ φ
- 8. 8 SOLO Optical Fiber – Ray Theory Graded-index Fiber (continue – 2( ( ) ( ) ( ) ( ) ( ) ( ) ( ) r rd rnd z sd zd rn sd d sd d sd rd rn sd d rn sd d r sd d rnr sd rd rn sd d sd rd rn sd d 11121 2 ray = + + + − = φ φφφ From this equation we obtain the following three equations: ( ) ( ) ( ) rd rnd sd d rnr sd rd rn sd d = − 2 φ ( ) ( ) 02 =+ sd d sd rd r rn sd d rn sd d φφ ( ) 0= sd zd rn sd d ( ) ( ) 022 =+ sd d sd rd rrn sd d rn sd d r φφ 2 r× ( ) 02 = sd d rnr sd d φ ( ) const sd zd rn == β ( ) .2 constl sd d rnr == ρ φ Integration Integration where: l,β -dimensionless constants (ray invariants( to be defined ρ -radius of the boundary between core and cladding By integrating the last two equation we obtain: (1) (2) (3) (3’) (2’)
- 9. 9 ( ) ( ) zrn sd zd rn θβ cos== SOLO Optical Fiber – Ray Theory Graded-index Fiber (continue – 3( We found that the ray propagation vector is Skew ray in core of fiber φ φ1 r1 z1 Q P zrs zzz 1cos1cossin1sinsin1 θφθθθθ φφ ++= Core Q' axis Core axis Cladding zθ s sd rd 1:ray = φ φ1 φθ r r1 φ1inner caustic outer caustic s1 z1 zθ ( )rnncore = sz sd zd sd d rr sd rd sd rd 1111 ray =++= φ φ ( )rnsd zd β = ( )rnr l sd d 2 ρφ = ( ) ( ) sd rd z rnrnr l r sd rd s ray 1111 =++= β φ ρ ( ) sd rd zrs zz ray 1cos1cos1sinsin1 =++= θφθθθ φφ Let write also as a function of two geometric parameterss1 φθθ ,z φθ -skew angle zθ -angle between ands1 z1 ( )rnr l z ρ θθ φ =cossin ( ) φθθ ρ cossin zrn r l = (3’) (2’)
- 10. 10 ( ) ( ) zrn sd zd rn θβ cos== SOLO Optical Fiber – Ray Theory Graded-index Fiber (continue – 4( We found φθ r r1 φ1inner caustic intesects ray path outer caustic intersects ray path 0=φθ 0=φθ The skew rays take a helical path, as seen from the cross-section figure. ( ) φθθ ρ cossin zrn r l = ( ) ( ) ( ) ( ) 22222 cossin cos β ρ θ ρ θ ρ θφ − = − == rn l rrnrn l rrn l r z z ( ) ( ) 0== ocic rr φφ θθ A particular family of skew ray will not come closer to the fiber axis than the inner caustic cylindrical surface of radius ric and further from the axis than the outer caustic cylindrical surface of radius roc. From the figure we can see that at the intersection of ray path with the caustic surface Therefore the caustic radiuses can be found by solving: ( ) ( ) 10cos 22 === − φ θ β ρ rn l r or ( ) ( ) 0: 2 2 222 =−−= r lrnrg ρ β ( ) ( ) 0== ocic rgrg
- 11. 11 SOLO Optical Fiber – Ray Theory Graded-index Fiber (continue – 5( We obtained: ( )rnsd zd β = ( )rnr l sd d 2 ρφ = ( ) zd d rnzd d sd zd sd d β == ( ) ( ) ( ) ( ) ( ) ( ) ( )rn rd rnd rnr l rnr zd rd rn rn zd d rn ×= − 2 2 ρββ ( ) 2 2 3 2 2 2 2 2 2 1 rd rnd r l zd rd =− ρ β Define: zd rd r =:' rd rd r zd rd rd d zd rd zd rd zd d zd rd ' '2 2 = = = ( ) 2 2 3 2 22 2 1 ' rd rnd r l rd dr r =− ρ β Integration ( ) constrn r l zd rd +=+ 2 2 2 2 2 2 2 1 2 1 2 1 ρ β ( ) const sd zd rn == β(3’) ( ) .2 constl sd d rnr == ρ φ (2’) ( ) ( ) ( ) rd rnd sd d rnr sd rd rn sd d = − 2 φ (1) ( ) ( ) 2 2 2 222 2 2 2 β ρ ββ +⋅+−−= const r lrn zd rd rg
- 12. 12 SOLO Optical Fiber – Ray Theory Graded-index Fiber (continue – 6( We obtained: ( ) ( ) 2 2 2 222 2 2 2 β ρ ββ +⋅+−−= const r lrn zd rd rg φθ r r1 φ1inner caustic intesects ray path outer caustic intersects ray path 0=φθ 0=φθ To determine the constant we use the fact that at the caustic we have therefore ( ) ( ) 0&0 2 2 222 =−−== r lrnrg zd rd ρ β 02 2 =+⋅ βconst Finally we obtain the ray path equation: ( ) ( ) 2 2 222 2 2 : r lrnrg zd rd ρ ββ −−== Since a ray path exists only in the regions where0 2 2 ≥ zd rd β ( ) 0>rg
- 13. 13 SOLO Optical Fiber – Ray Theory Graded-index Fiber (continue – 7( Analysis of: ( ) ( ) 2 2 222 2 2 : r lrnrg zd rd ρ ββ −−== A ray path exists only in the regions where ( ) 0>rg 1.Bounded rays The rays are bounded in the core region iff: g (r)>0 for ric<r < roc and g (r)<0 for r ≥ ρ rρ ocr icr 2 2 2 r l ρ cladding core 0≠l ( )rg skew ray β<cladding n( ) ociccore rrrrn ≤≤> β ( ) ociccorecladding rrrrnn ≤≤<< β rρ ocr 0=l cladding core ( )rg meridional ray
- 14. 14 SOLO Optical Fiber – Ray Theory Graded-index Fiber (continue – 8( Analysis of: ( ) ( ) 2 2 222 2 2 : r lrnrg zd rd ρ ββ −−== A ray path exists only in the regions where ( ) 0>rg 2.Refracted rays The rays are refracted from the core in the cladding region iff: g (r)>0 for r ≥ ρ rρ icr 2 2 2 r l ρ cladding core 0≠l ( )rg skew ray 222 lncladding +> β
- 15. 15 SOLO Optical Fiber – Ray Theory Graded-index Fiber (continue – 9( Analysis of: ( ) ( ) 2 2 222 2 2 : r lrnrg zd rd ρ ββ −−== A ray path exists only in the regions where ( ) 0>rg 3.Tunneling rays The rays escape in the cladding region iff: g (r)<0 for ρ <r<rrad and g (r)>0 for r ≥ rrad 222 lncladding +< β rρ ocr ic r 2 2 2 r l ρ cladding core 0≠l ( )rg skew ray radr β>cladding n 22 lncladding +<< ββ ( ) 02 2 222 =−−= rad claddingrade r lnrg ρ β 22 β ρ − = cladding rad n l r The energy leaks from the core to the cladding region.
- 16. 16 For a step-index core fiber ncore = constant. SOLO Optical Fiber – Ray Theory Core axis Cladding Skew ray in core of fiber z θ φθ s1 φ1 r1 z1 ρ Q P zrrs zzz 1cos1cossin1sinsin1 θθθθθ φφ ++= ρ φθ Core P Q Q' axis P Q' ρ φθρ sin2' =PQ φθ φθ icr φ θρ cos=ic r φθ inner caustic .constnn corecladding =< Step-index Fiber ( ) ( ) zrn sd zd rn θβ cos== ( ) φθθ ρ cossin zrn r l = ( ) ( ) 2 2 222 2 2 : r lrnrg zd rd ρ ββ −−== ( ) ≥= <= = ρ ρ rconstn rconstn rn cladding core 2 1
- 17. 17 SOLO Optical Fiber – Ray Theory Step-index Fiber (continue – 7( Analysis of: ( ) ( ) 2 2 222 2 2 : r lrnrg zd rd ρ ββ −−== A ray path exists only in the regions where ( ) 0>rg 1.Bounded rays The rays are bounded in the core region iff: g (r)>0 for r = ρ- ε and g (r)<0 for r = ρ+ε β<cladding nβ>core n corecladding nn << β rρ 22 β ρ − = core ic n l r 2 2 2 r l ρ claddingcore 0≠l ( )rg skew ray 22 β−core n 22 β−cladding n corenn = claddingnn = 0222 >−−= lng core β 0222 <−−= lng cladding β rρ 0=l claddingcore ( )rg meridional ray 022 <−= βcladdingng 022 >−= βcore ng corenn = claddingnn = ( ) ≥= <= = ρ ρ rconstn rconstn rn cladding core 2 1 ( ) 0=ic rg φ θθρ θβ θρ β ρ φ cos cossin cos22 zcore zcore nl n core ic n l r = = = − = P Q' ρ φθρsin2' =PQ φθ φθ icr φθρ cos=icr φθ inner caustic
- 18. 18 SOLO Optical Fiber – Ray Theory Step-index Fiber (continue – 8( Analysis of: ( ) ( ) 2 2 222 2 2 : r lrnrg zd rd ρ ββ −−== A ray path exists only in the regions where ( ) 0>rg 2.Refracted rays The rays are refracted from the core in the cladding region iff: g (r)>0 for r ≥ ρ 22 lncladding +> β ( ) ≥= <= = ρ ρ rconstn rconstn rn cladding core 2 1 rρ 22 β ρ − = core ic n l r 2 2 2 r l ρ claddingcore 0≠l ( )rg skew ray 22 β−coren 22 β−cladding n corenn = claddingnn = 0222 >−−= lng core β 0222 >−−= lng cladding β
- 19. 19 SOLO Optical Fiber – Ray Theory Step-index Fiber (continue – 9( Analysis of: ( ) ( ) 2 2 222 2 2 : r lrnrg zd rd ρ ββ −−== A ray path exists only in the regions where ( ) 0>rg 3.Tunneling rays The rays escape in the cladding region iff: g (r)<0 for ρ <r<rrad and g (r)>0 for r ≥ rrad 222 lncladding +< β β>cladding n 22 lncladding +<< ββ ( ) 02 2 222 =−−= rad claddingrade r lnrg ρ β 22 β ρ − = cladding rad n l r The energy leaks from the core to the cladding region. ( ) ≥= <= = ρ ρ rconstn rconstn rn cladding core 2 1 rρ 22 β ρ − = core ic n l r 2 2 2 r l ρ claddingcore 0≠l ( )rg skew ray 22 β−core n 22 β−claddingn core nn = cladding nn = 22 β ρ − = cladding rad n l r 0222 >−− lncore β 0222 <−− lncladding β
- 20. 20 For a step-index core fiber ncore = constant. SOLO Optical Fiber – Ray Theory P Q' ρ φθρ sin2' =PQ φθ φθ ic r φθρ cos=ic r φθ inner caustic Step-index Fiber ( ) ( ) 2 2 222 2 2 : r lrnrg zd rd ρ ββ −−== ( ) ≥= <= = ρ ρ rconstn rconstn rn cladding core 2 1 rρ 22 β ρ − = core ic n l r 2 2 2 r l ρ claddingcore 0≠l ( )rg skew ray 22 β−core n 22 β−cladding n corenn = claddingnn = 0222 >−−= lng core β 0222 <−−= lng cladding β rρ 0=l claddingcore ( )rg meridional ray 022 <−= βcladdingng 022 >−= βcoreng corenn = claddingnn = corecladding nn << β rρ 22 β ρ − = core ic n l r 2 2 2 r l ρ claddingcore 0≠l ( )rg skew ray 22 β−core n 22 β−claddingn corenn = claddingnn = 0222 >−−= lng core β 0222 >−−= lng cladding β rρ 22 β ρ − = core ic n l r 2 2 2 r l ρ claddingcore 0≠l ( )rg skew ray 22 β−coren 22 β−claddingn core nn = claddingnn = 22 β ρ − = cladding rad n l r 0222 >−− lncore β 0222 <−− lncladding β 1.Bounded rays 2.Refracted rays 222 lncladding +> β 3.Tunneling rays 22 lncladding +<< ββ
- 21. 21
- 22. 22 SOLO References C.C. Davis, “Laser and Electro-Optics”, Cambridge University Press, 1996, OPTICS S. Hermelin, “Foundation of Geometrical Optics”
- 23. January 9, 2015 23 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA

- S. Hermelin, “Foundation of Geometrical Optics”