Introduction to Nonlinear Optics. Simon-Pierre Gorza Marc Haelterman ULB - OptIQ

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1 Inroducion o Nonlinear Opics Simon-Pierre Gorza Marc Haelerman ULB - OpIQ

2 Inroducion o Nonlinear Opics Ouline - Inroducion - Polarizaion - Second-order nonlineariy - Second harmonic generaion - Phase maching - Parameric amplificaion - Third-order nonlineariy - Kerr effec - Self- and cross-phase modulaion - Four-wave mixing

3 3 Inroducion o Nonlinear Opics Ouline - Resonan nonlineariies - Lorenz oscillaor and -level model - Kramers-Krönig relaions - Oher nonlinear processes and maerials - Semiconducor nonlineariies - Cascading - Phoorefracive effec - Molecular reorienaion - Raman scaering - Device: figure of meri - Conclusions

4 4 Inroducion Nonlinear : Ligh-maer ineracions Maxwell s equaions + aoms

5 5 Maxwell s equaions = ro =! "B " ro B =! µ " " + µ J e e # ro ro =!" µ # - µ #J # J =!en " " Source erm Curren densiy : bound elecrons moion r (, ) $! "# µ $ = " µ en $ $ Polarizaion: P! " en! " 1 c # # = µ # P # = ( ) P =P ( ) =?

6 6 Ligh-induced polarizaion: a classical viewpoin Harmonic oscillaor F ex =!e e P =!en = Ne "! " 1 # c # = µ # P! " 1+ # $ # c $ = n = 1+ χ Ub α α d U b d χ = d =! e " e Ne d ε U b U b α U( ) = d F r =!" = e P! " # e =1.6 1!19 C! = 9 1 "1 C / (Vm) 9 3 N 1 m Ub 1 ev d A suscepibiliy χ = χ 3 Ne εα

7 7 Second-order nonlineariies Asymmeric elecron binding P =!en e F ex =!e A - D + P! U b U( ) α β = P! e!e F r =!"! # = e + P! " 1 # c # = µ # P # Source erms a, and

8 8 Nonlinear polarizaion U b U( ) α β = Linear: L =! e " Nonlinear:! " e # " $ # L e F r =!"! # = e! " e # " $ e # d =? P =!en = Ne " + # Ne3 " 3 P NL Second-order suscepibiliy P! " # +" # () χ () = Ne ε 3 β 3 α F r = α β = d when α = d β Ub = αd 6 α = β = 3 3 αα βα α Ub = U( d) = = 3 β 3 β 6β 6U b d 6U b 3 d χ () Ne d ε U b d =. nm () χ 1 pm/v χ () GaAs 1 pm/v

9 A D 9 Requiremens for frequency generaion 1. Non cenrosymmeric medium Alignmen needed β A D P NL = - Langmuir-Blodge - Crysals Ga D A Ga As Ga Ga ψ e U. Phase maching P NL =! " () = cos[k(!)z "!] P NL =! " () cos [k(#)z $#] P NL =! " () cos[k(#)z $ #]+! " () cos[ k( ) z] cos[ k( ) z] zc = π / Δk Δ k = k( ) k( ) z P! P! P zc = π / Δk z

10 1 Phase-maching echniques Δ k = k( ) k( ) =? 1. Birefringence OA n( ) n( ) = n( ) = n( ) c c n o θ n o (!) no( )! pm n e ( ) = n ( ) o e o θ ne no! () χ z

11 11. Quasi-phase maching 111 Ga P Ga Ga As Ga P Ga As Ga Ga P! 111 Ga P! zc = π / Δk cos[ k( ) z] cos[ k( ) z] zc z = π / Δ k z = π / Δk z c

12 1 Semiconducors: engineerable maerials Illusraion Al Ga x Muliple quanum wells: (1 x) As χ () Ne d ε U b ψ e g ( x) () Bulk Scale effec: A nm χ 1 pm/v χ () MQW 1 nm/v d = few nm Quasi-phase maching (surface emission) zc = π / Δk!

13 13 Parameric amplificaion P NL =! " () () χ Second harmonic generaion () χ 1 Phase-mached ineracion k( ) = k( ) Specrum = e!i" +#e!i" $ + * e i" +# * e i " $ = e!i4" + + #e!i("+" $ ) + # * e!i("!" $ ) + c.c + O() z k(!) " k(!#) = k(! "!#) Le: ʹ = P! NL = " # () 1 P NL εχ ε () * =! P NL!!

14 14 Third-order nonlineariy - Cenrosymmeric media U b U( ) α γ = P =!en =? e F r =!"!# 3 = e P!! " 1 # c # = µ # P # Source erms a and 3 P 3!

15 15 Nonlinear polarizaion U b U( ) α γ = Linear: L =! e " Nonlinear:! " e # " $ # 3 L d =? e F r =!"!# 3 = e! " e e3 +$ # # 3 4 P =!en = Ne "! # Ne4 " 4 3 P NL Third-order suscepibiliy P! " # +" # (3) 3! (3) = " Ne4 # $ % 4 F r = for α 3 = γ ( = ±d)! = "# / d! = 4U b d! = " 4U b d 4! (3) = Ne4 d 4 64" U b 3

16 16 Opical Kerr effec P =!en P NL =! " (3) 3 = cos[k(!)z "!] % 3 P NL =! " (3) 3 4 cos[k(#)z $#]+ 1 ( ' cos[3k(#)z $ 3#] * & 4 ) P! P NL = 3 4! " (3) P 3! P =! " + 3 4! " # (3) =! " " & % (3) ( $ ' 3 χʹ = χ + χ 4 (3) 3 3 nʹ = 1+ χʹ = 1+ χ+ χ = n + χ 4 4 (3) (3) n n 3 n 3 ʹ = (3) (3) χ χ 4 n 8 n I nε c [ ] I = W/ m nʹ = n+ ni n 3 m (3) = χ 4n εc W

17 17 Self- and cross-phase modulaion (3) Kerr effec: degenerae χ process P NL =! " (3) 3 = e!i" + * e i" = e!i" + * e i" + 3 = 3 e!i3" + * e i" + e!i" + e!i" + *3 e i3" + * e i" P! NL = P! NL e "i! + P NL! P " NL e #i"! * e i! (3) P 3 NL = εχ P NL = εχ (3) e e i1 i 1 1 = + 1 * i i * i i = e e e e P NL i 1 i 1 i * i 1 i = e e e e e i i i1 * i i e + e + e + e e!+! 1 = " # (3) $ ' 1 + %& () 1 n!+! 1 = n!+! 1 + n " # I 1 + I $ CPM % SPM =!+! = " # (3) $ ' + 1 %& () n!+! = n!+! + n " CPM # I + I $ 1 % Polarizaion 7 P NL SPM =

18 18 = e!i" + * e i" Four-wave mixing! P NL = " # (3) = 1 e!i(" 1!" ) + e!i("!" ) New frequencies = 1 1 e!i(" 1!" ) + 1 e!i(" 1!" ) + 1 e!i("!" ) + 1 * e!i" 1 e i" e!i" + e!i("!" ) + 1 e!i("!" ) + 1 e!i(" 1!" ) + 1 * e!i" e i" 1 e!i" (! P 1 "! ) NL = # $ (3) * 1 Phase maching: j = j ik ( ) z j A e (! P "! ) 1 NL = # $ (3) * 1 (! P 1 "! ) NL = # $ (3) A 1 A * e i[k (! 1 )"k (! )]z Specrum ( 1 ) grows if: δ = 1 k( ) = k( ) k( ) 1 1 ( ) 1 1 n = n + n I + I δ + δ

19 19 Resonan nonlineariies P =!en = ψ rdr e ϕ ϕ 1 ψ = cϕ ( r) e + cϕ ( r) e i1 i 1 1 ψ = c ϕ ( r) + c ϕ ( r) 1 1 "! j = % j $ ' # $ &' = Re% c * 1 c "! * 1 (r )! (r ) rdr e #i$ a & ' ( i a + Re[ ccϕ ( r) ϕ ( r) e ] * * 1 1 #! a = " & 1 % ( $ ' ϕ ϕ 1 =? i!"! = # $ H + e x % &" P =? H in

20 The Lorenz oscillaor U ( ) = a +! +" a = # e " m a = m! $ # a % U b Phasor represenaion: = e!i" = xe!i" d d e x iγx+ a x= m x e / m = iγ a P =!en P = Ne P =! " m e!i" " a!"! i#"! = Ne m" 1 # a $# $ i%# = ( + )( ) a a a χ = ( ) a a Ne 1 ( a )/ γ + i mε γ 1 + ( ) / γ a a χ = χʹ + iχʹ ʹ n= 1+ χʹ + iχʹ ʹ n = nʹ + inʹ ʹ k = n = kʹ + iα c = Ae ikz e!i" = Ae i k# z e!$z e!i" loss α = nʹ ʹ c

21 1 χ = Ne 1 ( a )/ γ + i mε γ 1 + ( ) / γ a a χʹ ( ) χʹ = Ne 1 ( a )/ γ mε γ 1 + ( ) / γ a a χʹ ʹ ( ) a χʹ ʹ = Ne 1 1 mε γ 1 + ( ) / γ a a a Two-level model: = ψ r dr wih i!"! = # $ H + e x % &" ψ = cϕ ( r) e + cϕ ( r) e i1 i 1 1 χʹ Fig. / s 4Boyd p. 3 I I = I/ Is I/ I s = * * i a = Re c1c ϕ1( r ) ϕ( r )] rdr e P =!en P =! "! = " c # a $(# $# a ) / % + i 1+ (# $# a ) / % + / S χʹ ʹ Fig. I/ IBoyd p.3 s I/ I s = I/ I = 4 ( )/ γ a s

22 Nonlineariies in he wo-level sysem! = " c # a $(# $# a ) / % + i 1+ (# $# a ) / % + / S S =! 4µ 1 T 1. Saurable absorpion "!! = # c 1 on resonance = a : $ a 1+ ($ %$ a ) / & + I / I S α α α I S! =! 1 1+ I / I s I χʹ ʹ () χʹ ʹ ( I) = 1 + I / IS! = n "" # c $ 1 1 # c % '' 1

23 3 χʹ χʹ ʹ. Kerr-like nonlineariy Fig. / Boyd 4 p. 3 I I = s I/ Is ( )/ γ a I/ Is I/ I s = I/ I s = I/ I = 4 s! = " c # a $(# $# a ) / % + i 1+ (# $# a ) / % + I / I S "! = # c %($ %$ a ) / & $ a ($ %$ a ) / & +1+ I / I S "! = # $ c & % a (% #% a ) 1 1+ (1+ I / I S ) / [(% #% a ) / & ] far from resonance :! "! a # [ χʹ ʹ ] "! = # $ c & % a (% #% a ) (1# & (% #% a ) # I I ) % S ( a) IS = IS γ! (3) eff = " c $ # a (# %# a ) 1 I S # > a < a : : χ > (3) eff χ < (3) eff x x I( x) I( x) focusing defocusing

24 4 Kamers-Krönig relaions αc ( )/ γ + i χ = ε γ a a 1 + ( a) / + I/ IS ( ) χʹ ( ) = a χʹ ʹ ( ) γ Seady sae Dynamics Impulse response P =! " Causaliy χ() χ() H() FT +% P() =! & "( # $ )( $ )d $ #%!(") #!(") $ H(") χ() H () " $ # 1 1 H( ) = δ( ) + iπ P()! % ' & + 1 χʹ ʹ ( ʹ ) χʹ ( ) = d π ʹ ʹ + 1 χʹ ( ʹ ) χʹ ʹ ( ) = d π ʹ ʹ Two-level model: ( ) χʹ ( ) = a χʹ ʹ ( ) γ

25 5 + 1 χʹ ʹ ( ʹ ) χʹ ( ) = dʹ π ʹ + c α ( ) n( ) = 1 + π ʹ ʹ dʹ α = nʹ ʹ c Nonlinear suscepibiliies Saurable absorpion!(", I) = # c " a $(" $" a ) / % + i 1+ (" $" a ) / % + I / I S χʹ I/ I s = 4 χʹ ʹ I/ Is I/ Is I/ I s = I/ I s = I/ I = 4 ( )/ γ ( a ) χʹ (, I) = χʹ ʹ (, I) γ + 1 χʹ ʹ ( ʹ, I) χʹ (, I) = d π ʹ ʹ a s K-K : Kerr effec χ = χ + χ (1) (3) χʹ + iχʹ ʹ = χ + χ + iχ + iχ (1) (3) (1) (3) r r i i ʹ χ dʹ + (3) (3) 1 χi ( ) r ( ) = π ʹ n= n + ni n + c α( ) ( ) = π ʹ Two-phoon absorpion: α = α + α I ʹ dʹ 1

26 6 Oher nonlinear processes and maerials 1. Nonlineariies in semiconducors Absorpion α 1. Band-filling g! g n ʹ ( I) g k g! k + c α ( ʹ, I) n(, I) = 1 + dʹ π ʹ α (, I) = α (,) +Δα ( ) n(, I) = n(,) +Δn( ) + c Δα ( ʹ ) Δ n( ) = π ʹ dʹ α Δα n e (! I )! K-K :! Kerr-like effec m n! 1 "1!n max ".5 W Δn τ 1 ns 1 ps!

27 7. Thermal nonlineariy g α I T I T Δα λ g Δn λ Kerr-like effec I λ Δn max.1. τ 1 µ s

28 8 3. Two-phoon Kerr nonlineariy Two-phoon absorpion Perurbaion heory!! g α! (" # g ) 3/ $ I k g / g! if! < g n Kramers-Krönig + c α( ) ( ) = π ʹ α ( ) = α( ) + α( )I ʹ dʹ n n TPA! / g GaAs n! 1.5"1 #17 m W Si (indirec) m n! 5"1 #18!n max " 1 #3! " 1 fs W

29 9. Cascading nonlineariies nd harmonic generaion: P NL =! " () () χ =! cos[k(!)z "!]+! cos[k(!)z "!] P NL =! " () { # cos [k(#)z $#]+ # cos [k(#)z $ #] } + εχ cos[ k( ) z ]cos[ k( ) z ] () { [ ] [ ]} + εχ k k z + k + k z () cos [ ( ) ( )] cos [ ( ) ( )] 3! " # ()! P! P 3! Conribuion o! "! () # $! () Δϕ NL ffecive Kerr effec

30 3 Opical recificaion: P NL =! " () () χ =! cos[k(!)z "!]+! " () # P NL =! " () # cos { [k(#)z $#]+ P NL! = " # ()! cos[k(!)z $!] = " # ()! P! = " #! + " # ()! = " $ # + # () & % '! () Conribuion o! "! () # $! () : ffecive Kerr effec + cos[ k( ) z ] P NL! () () χʹ = χ + χ nʹ = 1+ χʹ = 1+ χ+ χ = n + χ () χ nʹ n+ Pockels effec n } xample: LiNbO 3 () χ = 36 pm/v n!.1 "17 m / W! n GaAs

31 31 3. Phoorefracive effec Phoorefracive maerial: noncenrosymmeic insulaor or semiconducor wih rapping impuriy sies illuminaion absorpion charge exciaion charge ransfer and rapping _ i + space charge inernal field P NL =! " () =! cos[k(!)z "!]+ i Pockels effec Index variaion { P NL =! " () # cos [k(#)z $#]+ i () χ nʹ n+ i n + i cos[ k( ) z ]} P!

32 3 Illusraion: I( x) () χ nʹ n+ i n Δn( x) 1 + i _ x I( x ) Δn( x) x 1 1 ou x x wavefron amplificaion 1ou

33 33 4. Molecular reorienaion θ M + Tork: M = p! = p sin" ε U b Reorienaional nonlineariy Ne d χ =, n = 1+ χ p M + p θ M = p! = p sin" P! d P d n I m CS : n! 1 "19 W LC : 9 n 1 m W

34 34 5. Raman scaering Phoorefracive effec I( x ) Δn( x) 1 ρ 1 amplificaion ou i k ρ x x wavefron iρx iρx zxz (, ) = xxxz (, ) + iγ Δnxzxz (, ) (, ) x (,) = e 1 + e [ ρ = k sinθ] Temporal domain: 1 1 iβʹ ʹ 1 zz (, ) = z (, ) + iγ Δnzz (, ) (, ) (,) = e 1 + e I () Raman scaering : < response ime of he Kerr effec: Δn() 1 z 1ou 1ou ou i i 1

35 35 Maerial issue for devices : Figure of meri Nonlinear suscepibiliy versus absorpion and response ime n χʹ χʹ ʹ I/ I s = 4 I/ Is I/ I s = I/ I = 4 ( )/ γ n TPA a I/ Is I/ I s = s π Δ ϕ NL = ni ( z) dz I( z) = I() e αz λ Δϕ NL π n FOM... = I () λ α - Semiconducors (below bandgap): n =!1 "17 m / W # = 15 m "1 F.O.M.= 1!11 m / W - Silica (fibers): n =.45!1 " m / W # = 5!1 "5 m "1 F.O.M.= 3 1!9 m / W! / g

36 36 Conclusions Nonlinear opics: Simple physics wih an exremely rich specrum of complex behaviors of applied ineres Seemingly endless funcionaliies (room for creaiviy) Promising applicaions hanks o high speed and specificiy Obsacles: Opical nonlineariies are weak: ffors in maerial research and confinemen echniques.

37 37 References Beam shaping and conrol wih nonlinear opics, F. Kajzar and R. Reinisch, Plenum (New York, 1998). Chaper 1. A brief inroducion o nonlinear opics (in paricular, in semiconducors) Chaper. For he linear and nonlinear Kramers-Krönig relaions. Nonlinear opics in semiconducors II,. Garmire and A. Kos, Academic Press (San Diego, 1999). Chaper 1. Second-order nonlineariies in SC, he anharmonic poenial picure Nonlinear opics, R. W. Boyd, Academic Press (San Diego, 199). Chaper 4. Reorienaion nonlineariy. Chaper 5. Nonlineariies in he wo-level aomic model. Nonlinear opical effecs and maerials, P. Güner, Springer (Heidelberg, ). Chaper 1. Cascading nonlineariies Nonlinear phoonics, H. Gibbs, G. Khirova, and N. Peyghambarian, Springer-Verlag (Heidelberg, 199). Chaper 1. Semiconducor nonlineariies Maerials for nonlinear opics, R. Marder, J. Sohn, and Galen Sucky, ACS Symposium Series, (Boson,199). Chaper 1. Inroducion o nonlinear opics, he anharmonic poenial picure.

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