6η Διάλεξη Οπτικές ίνες

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1 6η Διάεξη Οπτικές ίνες Γ. Έηνας, Διάεξη 6, σε. Χρματική Διασπορά Γ. Έηνας, Διάεξη 6, σε. Pae

2 Intramodal disersion Otical sources are not monochromatic: otical ower wavelenth So we have to consider intramodal disersion time Γ. Έηνας, Διάεξη 6, σε. 3 Intramodal (chromatic) disersion Material Disersion: Occurs because refractive index is a nonlinear function of wavelenth (Fi. A). Grou velocity of a secific mode is a function of the refractive index, which causes the various sectral comonents of a iven mode to travel at different seeds accordin to their wavelenth Is sinificant in sinle-mode fibers, and is made worse by LEDs (which have a bier sectral width than laser diodes). Γ. Έηνας, Διάεξη 6, σε. 4 Pae

3 Intramodal (chromatic) disersion Fi.A Refractive index versus wavelenth for silica Γ. Έηνας, Διάεξη 6, σε. 5 Intramodal (chromatic) disersion 999 S.O. Kasa, Otoelectronics Inut Claddin v () Emitter Core v ( ) Very short liht ulse Outut Intensity Intensity Intensity Sectrum, Δ Sread, Δ t t t o t All excitation sources are inherently non-monochromatic and emit within a sectrum Δ, of wavelenths. Waves in the uide with different free sace wavelenths travel at different rou velocities due to the wavelenth deendence of n. The waves arrive at the end of the fiber at different times and hence result in a broadened outut ulse. Γ. Έηνας, Διάεξη 6, σε. 6 Pae 3

4 Intramodal (chromatic) disersion Waveuide Disersion: Occurs because only about 8% of the otical ower is confined to the core of a sinle-mode fiber. The liht roaatin in the claddin travels faster. It is insinificant in multimode fibers, whilst for sinle mode, material disersion is the dominant contribution. {See Fi.B}. Even if there was no material disersion, waveuide disersion would still exist; it is caused by the core-claddin structure of the fiber itself. Γ. Έηνας, Διάεξη 6, σε. 7 Waveuide Disersion With increasin wavelenth, more of the otical field (i.e. ower) enetrates into the claddin: y y Claddin > c > v Core v > v E(y) Claddin As more of the field is carried by the claddin, the rou velocity increases. Γ. Έηνας, Διάεξη 6, σε. 8 Pae 4

5 Disersion for SMFs Disersion (s/(nm.m)) - Fi.B: Disersion for a silica sinle-mode fiber - Γ. Έηνας, Διάεξη 6, σε. 9 Disersion Hence for sinle-mode fiber, minimum disersion is obtained at 3 nm However, minimum attenuation is at 55 nm. The units of disersion are: s/(nm.m) Pulse sreadin (in s) becomes worse with increasin distance (m) and with increasin sectral width of otical source (nm) D σ L σ D disersion, σ rms ulse sread, σ rms sectral width of source, L fiber lenth Γ. Έηνας, Διάεξη 6, σε. Pae 5

6 Phase & Grou Velocity For a disersive medium, such as sinle-mode fiber, the ulse shae will chane as it moves alon: Aart from some waveuide disersion, the main cause of the ulse sreadin is material disersion (nonlinear chane of n with ) couled with sources havin finite sectral width. Γ. Έηνας, Διάεξη 6, σε. Phase & Grou Velocity All otical sources (includin lasers) have a finite sectral width: Intensity (arbitrary units) Δ: sectral width, FWHM Each wavelenth will see a different value of refractive index, and so travel at different seeds: n Γ. Έηνας, Διάεξη 6, σε. Pae 6

7 Phase & Grou Velocity Althouh we mainly deal with wavelenth rather than frequency, for this discussion it will be more convenient to use frequency. We will also consider just two, very closely saced frequencies within the rou: Intensity (arbitrary units) δ - Γ. Έηνας, Διάεξη 6, σε. 3 Phase & Grou Velocity At any iven wavelenth, we can consider the liht to be an electromanetic wave whose electric field is a sinusoidal travellin wave (in the + z direction): E ( z, t) E cos ( z t) () π π v ( f ) T hase constant anular frequency hase velocity Γ. Έηνας, Διάεξη 6, σε. 4 Pae 7

8 Phase & Grou Velocity Hence if we tae the simlified icture of assumin that our otical source emits two closely saced frequencies and, the corresondin waves are: E E cos ( z ) E E cos ( z t) t The suerosition (addition) of these two waves ives the total waveform as: E T [ ( z t) + cos ( z )] E cos t Γ. Έηνας, Διάεξη 6, σε. 5 Phase & Grou Velocity Main use of the trionometric identity: we et: cos α + cos β cos ( α β ) cos ( α + E T E cos cos β ( ) z ( ) ( + ) z ( ) + t t ) () Γ. Έηνας, Διάεξη 6, σε. 6 Pae 8

9 Phase & Grou Velocity Let: ~ E E T E cos cos [ ( ) ( ) ] z t [ ( + ) z ( + ) t] E T [ z t] [ z t] ~ E cos cos (3) Γ. Έηνας, Διάεξη 6, σε. 7 Phase & Grou Velocity If the frequencies are closely saced, then: ( + ) ( ) In other words, >> and we can then thin of our resultant electric field E T as an amlitude-modulated wave: E T [ z t] [ z t] ~ E cos cos ENVELOPE Modulation frequency CARRIER Carrier frequency Γ. Έηνας, Διάεξη 6, σε. 8 Pae 9

10 Phase & Grou Velocity Hence E T tyically loos lie: Normalised field - Time Γ. Έηνας, Διάεξη 6, σε. 9 Phase & Grou Velocity E T [ z t] [ z t] ~ E cos cos ENVELOPE CARRIER Velocity of carrier is: v + + Phase velocity (4) Γ. Έηνας, Διάεξη 6, σε. Pae

11 Phase & Grou Velocity E T [ z t] [ z t] ~ E cos cos ENVELOPE CARRIER Velocity of enveloe is: v d d Grou velocity (5) Γ. Έηνας, Διάεξη 6, σε. Phase & Grou Velocity v Normalised field - Time v The sinal roaates at the rou velocity v. Note: The enveloe is not a hysical artefact; it reresents the maximum excursion of the wave amlitude. Γ. Έηνας, Διάεξη 6, σε. Pae

12 Phase & Grou Velocity v v From(4): and substitutin into (5): d v d + dv d Now, π/, hence: (6) d d π v v + d dv d d v v dv d (7) Γ. Έηνας, Διάεξη 6, σε. 3 Phase & Grou Velocity If the hase and rou velocities are equal, then the enveloe will travel at the same seed as the carrier wave, and there will be no disersion. From equation (7), this imlies that the hase velocity should not deend on wavelenth if we are to achieve disersionless transmission. v v v v no disersion disersion Γ. Έηνας, Διάεξη 6, σε. 4 Pae

13 Disersion Relation The lot between and is nown as the disersion relation. From (5), the radient of this curve will yield the rou velocity: x v x x v d d x x Γ. Έηνας, Διάεξη 6, σε. 5 Normal Disersion In normal disersion, the rou velocity is less than the hase velocity. v v v < v normal disersion Γ. Έηνας, Διάεξη 6, σε. 6 Pae 3

14 Anomalous Disersion In anomalous disersion, the rou velocity exceeds the hase velocity. v v v > v anomalous disersion Γ. Έηνας, Διάεξη 6, σε. 7 Grou Refractive Index In the context of otical fibers, imaine we have a fiber with core refractive index n. In this case, c v (8) n If we transmit a sread of wavelenths, then we can reard the resultin rou as encounterin a rou refractive index, and this is defined via: d c v (9) d n n c v () Γ. Έηνας, Διάεξη 6, σε. 8 Pae 4

15 Material Disersion We seen reviously that: Otical sources have a finite sectral width This leads to the definition of rou velocity Refractive index varies (nonlinearly) with wavelenth We will now examine how these two henomena combine to yield rou velocity disersion (material disersion). Γ. Έηνας, Διάεξη 6, σε. 9 Material Disersion We reviously considered just two, very closely saced frequencies within the rou emitted by an otical source such as a laser: Intensity (arbitrary units) δ - Γ. Έηνας, Διάεξη 6, σε. 3 Pae 5

16 Material Disersion Two closely saced frequencies: carrier ( + )/ -.5 enveloe ( - )/ Μάθημα HMY 455Συστήματα και Δίκτυα Επικοιννιών -.5 με Οπτικές Ίνες modulated waveform Γ. Έηνας, Διάεξη 6, σε. 3 Material Disersion If we consider the entire sectrum emitted by the source, we still obtain a modulated waveform, with a rou velocity etc. as before. Recall Fourier transform: f ( t) j t j t F ( ) e d F ( ) f ( t) e dt Time domain - δ π Frequency domain F() N.B. This reresents otical source sectrum; has a aussian rofile ea frequency + δ Γ. Έηνας, Διάεξη 6, σε. 3 Pae 6

17 Material Disersion We can thin of F() as bein equal to some sectrum G() which is identical in shae, but centred at instead of : G() F ) G ( ) ( F() - δ δ - δ + δ F ( ) j t f ( t) e dt π G ( ) π π ( t) e j ( ) t j t j t ( t) e e dt dt Γ. Έηνας, Διάεξη 6, σε. 33 Material Disersion Hence: Imulse resonse of: G() f ( t) ( t) e j t Corresonds to sinusoid at otical frequency ives: (t) Note: Fourier transform of a aussian ulse is also aussian in shae Γ. Έηνας, Διάεξη 6, σε. 34 Pae 7

18 Material Disersion In other words, the imulse resonse associated with the otical source taes on the form of a modulated waveacet: (t) t f (t) This waveacet reresents a ulse of liht emitted by the otical source, and it contains a rane of frequencies (i.e. wavelenths). We now need to examine what will haen to the rou velocity of this ulse as it roaates alon a fiber. Γ. Έηνας, Διάεξη 6, σε. 35 Material Disersion Consider an otical ulse launched into a sinle mode fiber. Due to the sectral width of the source, this ulse consists of a rou of wavelenths which travel at the rou velocity: otical ower v d d wavelenth distance Γ. Έηνας, Διάεξη 6, σε. 36 Pae 8

19 Material Disersion So the time taen for the waverou to travel a distance L down the fiber is iven by the rou delay τ : L d τ L () v d The hase velocity of the ea wavelenth is iven by: v c n Substitutin into () into (): τ d L n c () dn n + d c d (3) Γ. Έηνας, Διάεξη 6, σε. 37 Material Disersion Eqn. (3) shows that the rou delay er unit lenth deends on both n and dn/d. It is also deendent on the frequency. However, we refer to wor with wavelenth instead: n n instead of... Γ. Έηνας, Διάεξη 6, σε. 38 Pae 9

20 Material Disersion Given the inverse relationshi between frequency and wavelenth (c f /π), we miht exect that: τ L n c dn + d c n dn d but maybe we should rove this... Γ. Έηνας, Διάεξη 6, σε. 39 Material Disersion From (): n π n πf c T c n f Hence: π n (4) Comarin with (4) with (): n c πc (5) Γ. Έηνας, Διάεξη 6, σε. 4 Pae

21 Material Disersion Now, from (3), the rou delay er unit lenth can be re-exressed as: τ L n c Differentiatin (5) w.r.t. : dn + d n c πc d πc d τ L n c dn d + d d dn d (7) (6) Γ. Έηνας, Διάεξη 6, σε. 4 Material Disersion We reviously defined the rou refractive index: n c/v τ dn n c n (7) L d Now, nowin that n varies with wavelenth: dn d n n v v disersion In fact, n will also be wavelenth deendent, and the radient of the n vs. wavelenth curve is: dn d n (8) d d Γ. Έηνας, Διάεξη 6, σε. 4 Pae

22 Wavelenth deendence of n and n for fused silica At.3 μm, n has a oint of inflection, n is minimum, and the rou velocity is therefore maximum. Γ. Έηνας, Διάεξη 6, σε. 43 Material Disersion n n dn d n d d dn minimum, i.e. d.3 μm oint of inflection, i.e. d n d Γ. Έηνας, Διάεξη 6, σε. 44 Pae

23 Grou velocity disersion (GVD) We now that: An otical source emits a sread of wavelenths centred on. This can be reresented by a waveacet which travels at the rou velocity and therefore sees a rou index n. However, n and thus the rou velocity v and delay τ are all wavelenth deendent. Each different sectral comonent emitted by the source will travel at different rou velocities, and this GVD is the cause of material disersion. Γ. Έηνας, Διάεξη 6, σε. 45 Delay difference (er unit lenth) for a wavelenth δ away from the central wavelenth τ τ ( ) L L δ τ n τ ( + δ ) c δ L If the wavelenth difference is sufficiently small, we can nelect second-order terms in a Taylor series exansion to et: ( τ + δ + δ ) τ ( ) δ dτ L L d (9) Γ. Έηνας, Διάεξη 6, σε. 46 Pae 3

24 Material Disersion From (7): δτ L δ L τ L dτ d dn n c d δτ d n () L δ c d Delay difference (er unit lenth) for a wavelenth δ away from the central wavelenth Material disersion D mat Units: s/(nm.m) Γ. Έηνας, Διάεξη 6, σε. 47 Material Disersion D mat d n c d Γ. Έηνας, Διάεξη 6, σε. 48 Pae 4

25 Material Disersion D mat d n c d The actual sin of D mat does not matter (excet when dealin with solitons), it simly indicates which wavelenths are faster than others. In fact, the majority of boos lot - D mat versus wavelenth, and refer to D mat as the material disersion, as in the next slide. For a source with an rms sectral width of σ, the corresondin rms ulse sread after a fiber lenth L is iven by: σ mat D mat σ L () sread in time sread in wavelenth Γ. Έηνας, Διάεξη 6, σε. 49 Material Disersion Althouh D mat is zero at.3 μm, we should refer to this as the wavelenth of minimum disersion, not zero disersion. Why? Γ. Έηνας, Διάεξη 6, σε. 5 Pae 5

26 Προσέγγιση LP όπου x ρ w iβ z E Ae e A w β Πάτος Εύρος δέσμης Σταθερά διάδοσης Γ. Έηνας, Διάεξη 6, σε. 5 Σταθερά διάδοσης Η σταθερά διάδοσης εξαρτάται από τη συχνότητα. Με ανάπτυγμα σε σειρά Taylor βn n β ( ) (7) n n! n d β βn (8) n d Γ. Έηνας, Διάεξη 6, σε. 5 Pae 6

27 Διάδοση παμού Ένας παμός δημιουργείται στην είσοδο της ίνας E z E e β x x π E ( t,) f( t) (9) x Το φάσμα του παμού βρίσκεται με μετασχηματισμό Fourier it Ex(,) Ex( t,) e dt () Η διάδοση μιας συχνότητας περιγράφεται από τη σχέση i ( ) z x(, ) x(,) () Μετά τη διάδοση, το ΗΠ στο σημείο z βρίσκεται με αντίστροφο μετασχηματισμό Fourier it E ( t, z) E (, z) e d () Γ. Έηνας, Διάεξη 6, σε. 53 Προσέγγιση ης τάξης Κρατώ τους δύο πρώτους όρους της σειράς Taylor β ( ) β + β (3) (),(3) (9) iβz i( t βz) iβz Ex t z e Ex e d f t βz e π () (, ) (,) ( ) Χρόνος διάδοσης παμού L τ υ (4) όπου όρισα την ταχύτητα ομάδας υ β (5) Γ. Έηνας, Διάεξη 6, σε. 54 Pae 7

28 Διαφορική καθυστέρηση Ι Για παμό εύρους ζώνης Δ (4) (5) (8) dτ d L dβ Δ τ Δ Δ L Δ Lβ Δ (6) d d υ d όπου το β ονομάζεται παράμετρος διασποράς της ταχύτητας ομάδας Γ. Έηνας, Διάεξη 6, σε. 55 Διαφορική καθυστέρηση ΙΙ Εναακτική έκφραση, για εύρος ζώνης εκφρασμένο σε μ.κ. Δ (4) dτ d Δ τ Δ LΔ DLΔ d d υ (7) όπου όρισα την παράμετρο διασποράς d D d υ (8) Γ. Έηνας, Διάεξη 6, σε. 56 Pae 8

29 Σύνδεση D, β d dβ dβ d D d υ d d d (5) (8) (9 ) a π c d πc d (9 b) (9 c) (9 b ),(9 c ) π c (9 a) D β (3) Γ. Έηνας, Διάεξη 6, σε. 57 Μέγιστη επιτρεπτή διαφορική καθυστέρηση (5),(7) (4) Δτ T R DLΔ (3) b b Γ. Έηνας, Διάεξη 6, σε. 58 Pae 9

30 Αριθμητικό παράδειγμα Αριθμητικά δεδομένα Λύση (πούτροπο laser) ( μ ) D Δ 4 nm RL b s.3 m nm m Gb DΔ 5 m s δη. ένα σήμα.5 Gb/s πάει << m. Γ. Έηνας, Διάεξη 6, σε. 59 Μηχανισμοί χρματικής διασποράς Παράμετρος χρματικής διασποράς : D D + D M W (3) D D M W Διασπορά υικού Διασπορά κυματοδηγού Τα D, D έχουν αντίθετα πρόσημα και μηδενίζονται για.3 μm M W ZD Γ. Έηνας, Διάεξη 6, σε. 6 Pae 3

31 Βετίση χρματικής διασποράς Γ. Έηνας, Διάεξη 6, σε. 6 Συμπεράσματα Οι μονότροπες οπτικές ίνες επιτρέπουν τη μετάδοση σημάτν με ψηούς ρυθμούς μετάδοσης σε μεγάες αποστάσεις Η εξασθένιση κι η χρματική διασπορά θέτουν άν όρια στο ρυθμό σηματοδοσίας και την απόσταση μετάδοσης Οπτικοί ενισχυτές, ίνες με μικρή χρματική διασπορά κι εξιστές διασποράς χρησιμοποιούνται για την καταποέμηση τν παραπάν Γ. Έηνας, Διάεξη 6, σε. 6 Pae 3