( ) [ ] 1 v. ( ) E v. ( ) 2 and, thus, Equation [ IX-1 ] simplifies to. ( ) = 3 4 ε χ 3. ( ) + k 0 2 ε 0 ( ) = 0 [ IX-4 ]

Transcription

1 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 75 IX. OPTICAL PULSE PROPAGATION THE ELECTROMAGNETIC N ONLINEAR S CHRÖDINGER EQUATION : We begin ou discussion of optical pulse popagation 35 with a deiation of the nonlinea Schödinge (NLS) equation. To that end, we ecall Equations [ VII-3 ] and [ VII-3 ] fom the ealy lectue set entitled Nonlinea Optics I -- i.e. E,ω ( ) ( E (,ω)) + ω c ε 0 ε ω ( ) ( ) E (,ω) = µ 0 ω ε ω [ ( ) E (,ω )] = ( ) P NL ( ) P NL (,ω) [ IX-1 ] (,ω) [ IX- ] In this teatment we will confine ou attention to wae popagation in unifom, isotopic optical mateials -- iz., glass fibes. Fo such mateials, we can wite whee ε NL,ω whee k 0 = ω c. 36 ( ) = 3 4 ε χ 3 0 xxxx ( ) E,ω P NL ( ) E,ω (,ω) = ε NL,ω ( ) E (,ω) [ IX-3 ] ( ) and, thus, Equation [ IX-1 ] simplifies to ( ) + k 0 ε 0 ( ) [ ε( ω) +ε NL (,ω)] ( ) = 0 [ IX-4 ] E,ω To poceed, postulate that this nonlinea Helmholtz equation can be teated by sepaation of aiables methods. In paticula, we ae looking fo a time-localized solution (a 35 An excellent efeence on this subject is Goind P. Agawal s Nonlinea Fibe Optics, Academic Pess (1989) ISBN In this simplification, we hae taken E (,ω) = ε( ω) ε NL (,ω) E (,ω) ε NL (,ω) 0. [ ] 1 R. Victo Jones, Mach, 000

2 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 76 pulse) with a elatiely naow fequency spectum (o goup of fequencies centeed on a fequency ω ct. Thus, we assume a seoaation of aiables solution E,ω ( ) = F( x,y) ( ) exp ( i β ct z) [ IX-5 ] G z,ω ω ct whee β ct is a wae o popagation numbe to be associated with ω ct and, thus, Equation [ IX-4 ] becomes whee k = k 0 F x + F y + k β [ ] F G + G z i β G ct z + β β ct [ ] G F= 0 [ IX-6] [ ε ω (,ω)] ε 0. In the linea poblem β would be the sepaation ( ) +ε NL constant, but in this case we will need a bit moe elaboation. Neetheless, we shall assume that we can find a set of functions F x, y ( ) and alues β that satisfy the equation so that F x + F y + [ k β ] F= 0 [ IX-7a ] G z i β G ct z + β β ct [ ] G = 0 [ IX-7b ] To use petubation theoy, we fist educe Equation [ IX-7a ] to a solable linea poblem by witing k = k 0 [ ε ω (,ω)] ε 0 = n ω n ( ω)+n( ω) n( ω) ( ) +ε NL [ ] k 0 [ ( ) + n( ω) ] k 0 [ IX-8a ] β = [ β+ β] β +β β [ IX-8b ] R. Victo Jones, Mach, 000

3 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 77 whee n ω the linea equation,ω ( ) = ε NL( ) n ( ω) = 3 8 ( ω) n ω ( 3) χ xxxx ( ) E,ω ( ). Thus, to fist ode we need to sole F x + F y + n k 0 β [ ] F= 0. [ IX-9 ] which taken togethe with appopiate bounday conditions defines the linea eigenalue poblem fo popagation in the medium whee the functions F ae the eigenfunctions and the alues β ae the eigenfunctions. In the peious lectue set -- i.e., VIII. Guided Waes in Plana Stuctues -- we found a set of eigenfunctions and eigenalues appopiate to the dielectic-slab guidewae popagation Following an ealie discussion, we now pesume that G G << β z ct z -- i.e., we take the slowly ay amplitude o paaxial appoximation -- so that Equation [ IX-7b ] educes to G i β ct z + β β ct whee β β ct β β ct + β β and, thus, G i β ct z + β β ct [ ] [ ] G = 0 [ IX-10a ] G + β β G = 0 [ IX-10b ] Fo equation [ IX-7a ] to be completely satisfied in fist ode, we must hae R. Victo Jones, Mach, 000

4 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 78 β= n k 0 β = 3 k 0 8 β χ 3 xxxx n F x,y ( ) ( ) dx dy ( ) dx dy ( ) F ( x, y ) 4 dxdy F ( x, y) dxdy F x,y G z,ω ω ct [ IX-11 ] Since we ae assuming that popagating pulse has a elatiely naow fequency spectum, it is easonable to use a Taylo expansion aound ω ct fo β ω ( ) -- iz. β( ω) β ct + ( ω ω ct ) β (! ω ω ct) β + 1 ( 3! ω ω ct) 3 β 3 +K [ IX-1 ] whee β l = dl βω ( ) dω l. ω=ω c Nea the dispesion minimum in glass fibes (i.e. λ µm ) we may, to ey good appoximation, stop with the quadatic tem and wite β β ct β ct β β ct In this appoximation, Equation [ IX-10 ] becomes If we take G z + i [ ] = β ct ( ω ω ct ) β (! ω ω ct) β ( ω ω ct) β 1 + 1! ω ω ct ( ) β. [ IX-13 ] G + i β β G = 0. [ IX-14a ] β ct γ= β β= 3 k 0 ( 3) F x,y χ β ct 8β xxxx ct F x, y ( ) 4 dxdy ( ) dxdy [ IX-15 ] R. Victo Jones, Mach, 000

5 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 79 Equation [ IX-10 ] in the fequency domain educes to G z + i ( ω ω c)β 1 + 1! ω ω c ( ) β G = i γ G G. [ IX-14b ] which implies the following time domain equation fo the pulse enelope: G ( z,t) z G ( z,t) +β 1 i t β G ( z,t) + α t G z,t ( )= i γ G z,t ( ) G ( z,t) [ IX-16 ] The last tem on the left hand side has been added to incopoate the effects of aious possible loss mechanisms. Next we tansfom into a coodinate system which moes with the goup -- what might be called the sufe's coodinates of the pulse -- i.e., { t, z} { τ,ς} whee τ= t t 0 β 1 ( z z 0 )and ς= z z 0 = z β 1 1 t 0. In tem of these sufe s coodinates, Equation [ IX-16 ] becomes 37 G ( ς,τ) = α ς G ς,τ ( )+ i β G ( ς,τ) τ i γ G ς,τ ( ) G ( ς,τ) [ IX-17 ] Obiously, if we omit all of the tems on the ight hand side of this equation so that G ς,τ = 0, we would hae the ideal situation wheein a pulse of any shape ( ) ς popagates foee without changing shape at a elocity g = β the goup 37 Since z t G ( z,t) = ς G ( z,t) = ς G ( ς,τ) ς z + τ G ( ς,τ) ς t + τ G ( ς,τ) τ z = ς G ( ς,τ) τ t = ς G ( ς,τ) ( 1)+ τ G ( ς,τ) ( 0)+ τ 1 G ( ς,τ) ( g ) G ς,τ ( ) 1 ( ). R. Victo Jones, Mach, 000

6 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 80 elocity. In teating the less-than-ideal situation, we will initially neglect the effects of the loss (fist) tem and confine ou attention to the competing effects of the dispesion (second) and nonlinea (thid) tem. We cast the lossless esion of Equation [ IX-16 ] into a nomalized, standad fom by intoducing u ( Z,T) = G ς,τ ( ) P 0, Z = ς L D = ς β τ, T = τ τ whee τ is the width of the pulse and P 0 is its peak powe. Thus, we, at last, obtain the standad fom of the nonlinea Schödinge (NLS) equation i u( Z,T) sgn( β Z ) u( Z,T) whee N = L D L NL = γ P 0 τ β. T + N u Z,T ( ) u( Z,T) = 0 [ IX-18 ] PULSE S OLUTIONS OF LINEAR S CHRÖDINGER EQUATION ( ) U( z,t) and neglect the loss and nonlinea tems, ( ) satisfies the following diffeential equation: If in Equation [ IX-10 ] we set G z,t we see that U z, t Fo minimal dispesion -- iz. if b = 1! z U ( z, t 1 ) + g t U ( z, t ) = ib t U ( z,t ) [ IX-19 ] d β( ω ) 0 -- Equation [ IX-19 ] dω ω =ω 0 becomes z U( z, t) + 1 g t U( z, t) = 0 [ IX-0 ] R. Victo Jones, Mach, 000

7 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 81 which is the basic wae equation fo minimally dispesie media with the geneal solution U( z, t) = U( z g t) [ IX-1 ] whee g = d β( ω ) dω ω =ω 0 1 is the goup elocity of the pulse. When limitation to fist ode dispesion is an adequate appoximation, Equation [ IX-19 ] expessed in sufe s coodinates becomes whee b = β. b τ U ( ς,τ ) + i ς U ( ς,τ ) = 0 [ IX- ] Amazingly, this pulse dispesion equation is the, so called, paabolic equation that we saw ealie in connection with beam popagation -- iz. the paaxial wae popagation equation. Solution of Pulse Dispesion Equation Fo conenience, we estate hee Equation [ IX- } the fist-ode pulse dispesion equation -- iz. b τ U ( ς,τ ) + i ς U ( ς,τ ) = 0. Let us wite a Fouie tansfom fo this modulation in tems of "sufe time" -- i.e. R. Victo Jones, Mach, 000

8 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 8 whee ( ) = U( ς,ω) exp( i ω τ) dω U ς, τ U( ς,ω) = 1 π + [ IX-3a ] + U( ς, τ) exp( i ω τ) d τ. [ IX-3b ] Thus, the pulse dispesion equation educes to an odinay diffeential equation -- iz. i d dς U ( ς,ω ) = b ω U( ς,ω) [ IX-4 ] fo the Fouie tansfom and thus we hae the simple solution U( ς,ω) = U( 0,ω) exp( i b ω ς). [ IX-5 ] Thus, we see that the dispesion changes the phase of each spectal component of the pulse by an amount that depends on the fequency and the popagated distance. The geneal solution may be witten as whee ( ) = U( 0,ω) exp i ω τ b ω ς U ς, τ + [ ( )] dω [ IX-6a ] U( 0,ω) = 1 π + U( 0, t t 0 ) exp [ i ω ( t t 0 )] d( t t 0 ). [ IX-6b ] Let us suppose that we hae a "chiped" Gaussian at ς = 0 -- i.e. R. Victo Jones, Mach, 000

9 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 83 so that ( ) = exp U 0, t t 0 1 ic t t 0 τ 0 [ IX-7a ] U( 0,ω) = τ 0 π 1 ic ω ( ) exp τ 0 ( 1 ic) [ IX-7b ] whee C > 0 chaacteizes an "up-chip" and C < 0 a "down-chip" pulse. A Gaussian pulse with a fequency "down-chip" Ineting the tansfom, we see that R. Victo Jones, Mach, 000

10 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 84 U( ς,τ) = τ 0 τ 0 + i b ς 1 ic ( ) exp ( 1 ic) τ τ 0 + i bς 1 ic [ ( )] [ IX-8a ] o ationalizing U( ς,τ) = τ 0 τ 0 + i b 1 ic ( ) exp τ τ 0 1+ b C ς + 1+ b ς τ 0 τ 0 i C + b ς 1+C ( ) τ exp 0 τ τ 0 1+ b C ς + 1+ b ς τ 0 τ 0 [ IX-8b ] Hence, the pulse width boadening facto at a gien position z is gien by τ( ς) = 1+ bc ς + b ς τ 0 τ 0 τ 0 = 1+ C ς + ς L D L D [ IX-9] whee L D = τ 0 b. R. Victo Jones, Mach, 000

11 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 85 The spatial eolution of the pulse width of chiped Gaussian pulse (Fo "nomal dispesion" -- i.e. β ω >0) Pulse Solutions of Negligible Dispesion Nonlinea Schödinge Equation If goup elocity dispesion can be neglected, Equation [ IX-17 ] educes to G ( ς,τ) = α ς G ς,τ ( ) i γ G ς,τ and if we take G ς,τ ( ) U ς,τ ( ) P 0 exp ας ( )we obtain ( ) G ( ς,τ) [ IX-30 ] U ( ς,τ) -1 = i L ς NL exp ( ας) U( ς,τ) U ( ς,τ) [ IX-31 ] R. Victo Jones, Mach, 000

12 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 86 whee the chaacteistic nonlinea length is gien by L NL = ( γ P 0 ) 1. A solution to this equation is eadily obtain in the fom U ( ς,τ) = U ( 0,τ) exp i Φ NL ( ς, τ) [ ] [ IX-3a ] whee and ( ) = 1 exp( ας) [ ] 1 exp( ας) U ( 0,τ) [ IV-3b ] = [ ς eff L NL ] U ( 0,τ) Φ NL ς, τ [ ] α L NL ς eff = [ 1 exp ( ας) ] α 1 [ IX-3c ] This inteesting esult show that so called self-phase modulation o SPM gies ise to an intensity dependent phase shifted o chiped pulse which emains constant in shape as it popagates. The instantaneous optical fequency shift is gien by δω τ ( ) = Φ ς, τ NL( ) τ [ ] U ( 0, τ ) = ς eff L NL τ [ IV-33 ] Note that the pulse spectum is ed-shifted on the leading edge of a pulse and blueshifted on the tailing edge of the pulse. If we suppose the initial pulse to be a supe- Gaussian of mth-ode -- i.e., ( ) = exp U m 0,τ 1+ ic τ τ m [ IV-34 ] -- then the instantaneous optical fequency shift would be gien by δω τ ( ) = ς eff L NL m τ τ τ m-1 τ exp τ m [ IV-35 ] R. Victo Jones, Mach, 000

13 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 87 Supe-Gaussian Enelope Shapes SPM Induced Fequency Chip A Bief Discussion of Solitons We etun biefly to Equation [ IX-18 ] -- the standad fom of the nonlinea Schödinge (NLS) equation -- i.e., i u( Z,T) sgn( β Z ) u( Z,T) T + N u Z,T ( ) u( Z,T) = 0 [ IX-18 ] R. Victo Jones, Mach, 000

14 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 88 whee u ( Z,T) = G ς,τ ( ) P 0, Z = ς L D = ς β τ, T = τ τ and N = L D L NL = γ P 0 τ β. In soliton analysis the most common fom of NLS equation is the following: i U ( Z,T) Z + U ( Z,T ) + U ( Z,T) U ( Z,T) = 0 T [ IX-36 ] whee U ( Z,T) = N u ( Z,T) G ( ς,τ) γ τ β. If N 1 the following fundamental soliton will popagate undistoted fo an abitay distance: If U ( Z,T) = sech( T) exp( i Z ) [ IX-37 ] N the following second-ode soliton will popagate undistoted fo an abitay distance: U Z,T ( ) = 4 cosh ( 3 T ) + 3exp ( i 4 Z) cosh( T) cosh( 4 T) + 4 cosh( T) + 3 cos( 4 Z) U Z,T [ ] [ ] exp i Z ( ) [ IX-37 ] [ ] [ IX-37 ] [ ] ( ) = 16 cosh ( 3T) + 9cosh ( T) + 6 cos( 4 Z) cosh( 3 T) cosh( T) cosh( 4 T) + 4 cosh( T) + 3cos( 4 Z) R. Victo Jones, Mach, 000

15 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 89 TWO S OLITON COLLISION ILLUSTRATIONS The following ae time-lapsed illustation of the popagation and collision of two solitons -- plot hee is u( x, t) = 1 ( ) + cosh ( 4 x 64t ) ( ) + cosh( 3x 36t) 3 + 4cosh x 8t [ 3cosh x 8t ] R. Victo Jones, Mach, 000

16 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 90 R. Victo Jones, Mach, 000

17 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 91 R. Victo Jones, Mach, 000