v r 1 E β ; v r v r 2 , t t 2 , t t 1 , t 1 1 v 2 v (3) 2 ; v χ αβγδ r 3 dt 3 , t t 3 ) βγδ [ R 3 ] exp +i ω 3 [ ] τ 1 exp i k v [ ] χ αβγ , τ 1 dτ 3

Transcription

1 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 58 VII. NONLINEAR OPTICS -- CLASSICAL PICTURE: AN EXTENDED PHENOMENOLOGICAL MODEL OF POLARIZATION : As an introuction to the subject of nonlinear optical phenomena, we write, in the spirit of Equation [ I-4 ], the most general form of higher orer terms in the phenomenological electric fiel expansion of the polarization ensity (which may then be inserte in Equations [ I-3 ]) as P α (NL) ( r, t) = ε 0 βγ r r t t E β E γ r t () r t χ αβγ r r, t t ; r r, t t +ε 0 r t r t (3) r 3 t 3 χ αβγδ r r, t t ; r r, t t ; r [ VII- ] ( r 3, t t 3 ) βγδ r r r 3 t t t 3 E γ E δ +L. E β r, t r, t The wae ector frequency epenent secon thir orer susceptibilities are then efine as r 3, t 3 r, t r, t () χ αβγ, ω ; k, ω = R R τ τ R τ R τ exp i k exp i k [ R ] exp +i ω τ [ R ] exp +i ω τ [ ] χ αβγ [ ] () R, τ ; R, τ [ VII-a ] (3) χ αβγδ, ω ; k, ω ; k 3, ω 3 = R R R 3 exp i k R τ R τ R 3 τ 3 exp i k τ τ τ 3 [ R ] exp +i ω τ (3) χ αβγδ R, τ ; R, τ ; [ ] exp i R 3, τ 3 k 3 [ R ] exp +i ω τ [ R 3 ] exp +i ω 3 τ 3 [ ] [ ]. [ VII-b ] R. Victor Jones, February, 000

2 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 59 Thus, we may write quite generally 6 P α (NL) ( r, t) = ε 0 βγ k ω ω ω k ω exp i ( + k ) r [ ] exp i ω +ω [ t] () χ αβγ, ω ; k, ω E β,ω E γ k, ω +ε 0 k ω k ω k 3 ω 3 exp i ( + k + k ) r 3 βγδ k k 3 ω ω ω 3 χ αβγδ (3), ω ; k, ω ; k 3, ω 3 E β [ ] exp i ω +ω +ω 3, ω E γ k, ω E δ [ t] k 3,ω 3 +L.. [ VII-3 ] A SIMPLE CLASSICAL MODEL OF NONLINEAR OPTICAL RESPONSE A simple Lorentz-Due moel is often use in the literature as a aluable guie to the unersting of the frequency behaior of the nonlinear ielectric response. 7 We assume that the potential energy of a one-imensional nonlinear (anharmonic) oscillator may be written V( x) = V ( x) +V 3 ( x) +V 4 ( x) +L = M ω o x + 3 Ma x3 + 4 Mb x4 +L [ VII-4 ] Thus, the equation of motion of a particle moing in that potential becomes (See figures on next page) x +Γx +ω o x + a x + b x 3 +L= q M E ( t ) [ VII-5 ] 6 () χ αβγ must anish for any material that is inariant uner inersion, since both r r P E are ectors, are (3) thus o uner inersion symmetry. Note also that χ αβγδ for a gien material has the same transformation () (3) properties the elastic constants of that material. The nonzero elements of χ αβγ χ αβγδ for arious crystal symmetries are compile in Y.R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 984). 7 See, for example, N. Bloembergen, Nonlinear Optics (The Aance Book Program), Aison-Wesley (99), ISBN R. Victor Jones, February, 000

3 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 60 Terms in Anharmonic Potential Anharmonic Potential x 4 x x Displacement Displacement 0 We can analyze the response of the oscillator by exping the isplacement in powers of the electric fiel E t -- iz. where x (n ) t x( t) = x () ( t) + x () ( t)+ x (3) ( t)+l [ VII-6 ] is proportional to the nth power of the fiel E ( t). Inserting this expression, we obtain the following into Equation [ VII-5 ] equating like powers of E t hierarchy of equations: x () ( t) +Γx () ( t) +ω o x () ( t) = ( q M) E( t) [ VII-7a ] x () ( t) +Γ x () ( t)+ω o x () ( t)+ a x () t [ ] = 0 [ VII-7b ] x (3) ( t) +Γ x (3) ( t)+ω o x (3) ( t)+ a x () t M M x () t [ ] 3 = 0 [ VII-7c ] + b x () t R. Victor Jones, February, 000

4 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 6 In the frequency omain we see that ω x () ( ω) iω Γx () ( ω) +ω o x () ( ω) = ( q M) E ( ω) [ VII-8a ] or x () ( ω) = ( q M) E ( ω) [ ω o ω iω Γ] = ( q M) E( ω) F( ω 0,ω;Γ) [ VII-8b ] u i w a riing term in Equation [ VII-7b ] -- iz. where F u,; w x () t [ ]. 8 Thus, knowing x () ω x () ( t) +Γ Therefore, in the frequency omain x () ( t)+ω o x () t = a x () ( t), we may then consier [ ] [ VII-9 ] x () ( ω) = a F ( ω 0,ω; Γ) ω x () ω x () ( ω ω ) [ VII-0a ] which, in iew of Equation [ -8b ], becomes x () ( ω) = a ( q M) F [ ω 0,ω; Γ] ω F ω 0, ω ; Γ [ ; Γ] E ω [ ] F ω 0, ω ω E ( ω ω ).[ VII-0b ] We may treat the thir orer terms in a similar manner. We write Equation [ VII-7c ] as 8 Note that near resonance F( ω 0,ω; Γ) i x (3) ( t) +Γ x (3) ω i ( ω ω 0 )+Γ ( t)+ω o x (3) t [ ] = i = a x () ( t) x () t + ( Γ ) Γ i ω 0 ω ω ω 0 ω b x () t [ ] 3 [ VII- ] = i ω D ( ω ω; Γ 0 ) where D is the so calle complex Lorentzian. R. Victor Jones, February, 000

5 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 6 In the frequency omain x (3) ( ω) = a F ( ω 0,ω; Γ) b F ω 0,ω; Γ ω x () ω x () ( ω ω ) ω ω x () ω x () ( ω ) x () ( ω ω ω ). VII- ] Using Equations [ VII-8b ] [ VII-0b ], we obtain x (3) ( ω) = a ( q M) 3 F [ ω 0,ω; Γ] F ω 0, ω ; Γ [ ] F ω 0, [ ( ω ω ); Γ] E ω ω ω ω F ω 0,( ω ω ); Γ [ ] F ω 0, b ( q M) 3 F [ ω 0,ω; Γ] ω ω F ω 0, ω ; Γ F [ ω 0,( ω ω ω ); Γ]E ω [ ω ; Γ] [ ] F [ ω 0, ω ; Γ] E ( ω ) E ( ω ω ) E ( ω ) E ( ω ω ω ) [ VII-3 ] In particular, for an input E ( t) = E cos( ω t +ϕ )+ E cos( ω t +ϕ ) = E exp iω ( t ) + E [ exp ( +iω t) ] + E exp iω ( t ) + E exp ( +iω t) [ ] [ VII-4 ] a. Secon harmonic generation is ue to the terms x () ( ω ) = a ( q M) F [ ω 0,ω ;Γ] F [ ω 0,ω ;Γ] E ( ω ) [ VII-5a ] x () ( ω ) = a ( q M) F [ ω 0,ω ;Γ] F [ ω 0,ω ;Γ] E ( ω ) [ VII-5b ] b. Sum frequency generation is ue to the term R. Victor Jones, February, 000

6 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 63 [ ] x () ( ω +ω ) = a ( q M) F ω 0,( ω +ω );Γ F [ ω 0,ω ; Γ] F [ ω 0,ω ; Γ] E ( ω ) E ( ω ) c. Difference frequency generation is ue to the term [ ] x () ( ω ω ) = a ( q M) F ω 0,( ω ω );Γ F [ ω 0,ω ; Γ] F [ ω 0,ω ; Γ] E ( ω ) E ( ω ) [ VII-6a ] [ VII-6b ]. Optical rectification or c generation is ue to the terms x () ( 0) = a ( q M) F [ ω 0,0; Γ] F [ ω 0,ω ; Γ] E ( ω ) [ VII-7a ] x () ( 0) = a ( q M) F [ ω 0,0; Γ] F [ ω 0,ω ; Γ] E ( ω ) [ VII-7b ] e. Thir harmonic generation is ue to the terms { } x (3) ( 3ω ) = ( q M) 3 a F [ ω 0,ω ;Γ] b F [ ω 0,3ω ; Γ] F 3 ω 0,ω ;Γ [ ]E 3 ω [ VII-8a ] { } x (3) ( 3ω ) = ( q M) 3 a F [ ω 0,ω ;Γ] b F [ ω 0,3ω ; Γ] F 3 [ ω 0,ω ;Γ] E 3 ( ω ) [ VII-8b ] f. Intensity epenent propagation is ue to the terms { [ ] b} x (3) ( ω ) = ( q M) 3 a F ω 0,ω ;Γ F [ ω 0,ω ; Γ] F [ ω 0,ω ; Γ] E ( ω ) E ( ω ) [ VII-9a ] R. Victor Jones, February, 000

7 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 64 { [ ] b} x (3) ( ω ) = ( q M) 3 a F ω 0,ω ;Γ F [ ω 0,ω ; Γ] F [ ω 0,ω ; Γ] E ( ω ) E ( ω ) [ VII-9b ] g. Raman generation (inelastic scattering) inoles terms like { } x (3) ( ω ω ) = ( q M) 3 a F [ ω 0,( ω ω ); Γ]+a F [ ω 0,ω ; Γ] b F [ ω 0,( ω ω );Γ]F ω 0,ω ; Γ [ ] F [ ω 0,ω ; Γ] E ( ω ) E ( ω ) [ VII-0 ] Note that, accoring to this simple anharmonic oscillator moel, Raman generation may be enhance by a resonance at a frequency ( ω ω )! Also note that, for this moel (see Equation [ VII-3a ] aboe), the ratio χ () ω χ ( ω) χ () ( ω ω ) χ () ( ω ) = a M N q 3 [ VII- ] is a constant inepenent of frequency! This obseration is consistent with the famous empirical Miller Rule which eclares that the ratio ijk = ( ω 3 =ω +ω ) χ () jj ( ω ) χ () kk ( ω ) () χ iijk χ () ii ω 3 has only a weak ispersion is almost a constant for a wie range of materials! S ECOND HARMONIC GENERATION -- PERTURBATION ANALYSIS We may write the nonlinear, macroscopic Maxwell equations in the form E r,ω [ E ( r,ω)] + ω c ε 0 = µ 0 ω ε E r,ω P NL [ VII- ] ( r,ω) [ VII-3 ] R. Victor Jones, February, 000

8 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 65 ε E r,ω [ ] = Suppose that we hae an input riing or pump fiel E r,ω = ˆ e E ω P NL exp i ( z ω t) ( r,ω) [ VII-4 ] +c.c. [ VII-5 ] Then the components of the polarization with frequency ω are gien by P ( NL) ( z,ω = ω ) = E ( ω ) :ˆ ˆ χ e e [ ] + c.c. [ VII- 6 ] exp i z ω t It is a conenience to resole this P ( NL) the resultant secon harmonic fiel into longituinal transerse components -- iz. P ( NL) ( NL) ( z,ω ) = ˆ z P ( z,ω ) + P NL. [ VII-7a ] z,ω E ( z,ω ) = ˆ z E ( z,ω )+ E ( z,ω ) [ VII-7b ] Therefore, we may write the ω or secon harmonic components of Equations [ VII-3 ] [ VII-4 ] E z z,ω + ω c ε ω E ( z,ω ) +µ 0 ω ( NL) P ( z,ω ) ε 0 + ω ε ω E c ε ( z,ω )+µ 0 ω NL P 0 z,ω [ VII-8a ] ˆ z = 0 ( NL) { E ( z,ω )+ P ( z,ω )} = 0 [ VII-8b ] z ε ω To satisfy these equations two conitions must hol -- iz. R. Victor Jones, February, 000

9 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 66 E ( z,ω )= ε ω P ( NL) [ VII-9a ] z,ω z + k E ( z,ω ) = µ 0 ω ( NL) P [ VII-9b ] z,ω where We now write E z,ω function of z -- iz. z E z,ω = ˆ k ω c = ˆ E z,ω e ε ( ω ) ε 0 = ω µ 0 ε ( ω ) [ VII-30 ] exp ( i k z) where E ( z,ω ) is a slowly arying e k E( z,ω ) ik z E ( z,ω ) z E z,ω ( ) exp ( ik z ) [ VII-3 ] k E( z,ω ) i k z E ( z,ω ) exp ( i k z ) so that Equation [ VII-9b ] may be written z E ( z,ω ) = i µ 0 ω k ˆ e : = iµ ω 0 E ω k ˆ P NL : e exp ( ik z) χ z,ω :ˆ ˆ e e exp i kz [ VII-3 ] where k k = ω c [ n FH ( ω ) n SH ( ω )]. R. Victor Jones, February, 000

10 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 67 If we assume that the riing fiel stays constant, we can irectly integrate Equation [ VII-3 ] to obtain the exceeingly famous important equation for the spatial ariation of the secon harmonic fiel -- iz. E ( z,ω ) = z iµ 0 ω e ˆ : χ : ˆ 4 k e ˆ exp ( i k z ) E ω e sinc [ kz ].[ VII-33 ] MAKER 9 FRINGES SECOND HARMONIC INTENSITY (RELATIVE INTENSITY) LASER BEAM z = / cos θ SECOND HARMONIC ANGULAR ROTATION IN DEGREES S ECOND H ARMONIC G ENERATION - C OUPLED W AVE A NALYSIS When the process of secon harmonic generation takes place uner conitions of perfect phase match -- i.e. k = -- the perturbation result breaks own if the path is sufficiently long. Uner these circumstances the pump beam will be eplete as the secon harmonic grows a solution of Equation [ VII-3 ] must take into account the spatial ariation of E ( ω ). To that en we assume perfect phase matching 9 P.D. Maker, R.W. Terhune, N. Nisenoff, C.M. Saage, Phys. Re. Lett., 8, (965). R. Victor Jones, February, 000

11 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 68 rewrite Equation [ VII-3 ] as z E z,ω = i µ ω 0 : ˆ e :ˆ ˆ χ e e [ VII-34 ] Ε z,ω Of course, as the secon harmonic grows Equation [ VII-3 ] tells us that a nonlinear polarization at the pump frequency is generate -- iz. P (NL) ( z,ω ) = E( z,ω ) E ( z,ω ) χ [ :ˆ e e ˆ + χ :ˆ e () ˆ e ] [ VII-35 ] Repeating the arguments associate with Equations [ VII-8 ] through [ VII-3 ] we may write z E ( z,ω ) = i µ k 0 ω ( NL) P [ VII-36 ] z,ω R. Victor Jones, February, 000

12 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 69 Combining these two equations, we fin an equation goerning the spatial eolution of E ( z,ω ) -- iz. z E z,ω = i µ ω 0 [ ˆ e : χ : e ˆ ˆ e + e ˆ : χ :ˆ e e ˆ ] E z,ω E ( z,ω ). [ VII-37 ] Equations [ VII-34 ] [ VII-37 ] are then the couple ifferential equation which escribe the coupling of the first secon harmonic fiels. Hling all the "ectorness" in these two equation woul obscure important issues. Thus, we consier a pair of somewhat less complicate equations which incorporate the essence of the problem -- iz. z E z,ω z E z,ω = i µ ω 0 = i µ ω 0 To sole these equation we first write E z,ω with the bounary conitions f 0 to the imensionless form χ E z,ω k χ E z,ω k [ VII-38a ] E ( z,ω ) [ VII-38b ] = E ( ω ) f ( z) E ( z,ω ) = E ( ω ) f ( z) = f ( 0) = 0. Thus, the couple equations reuce ξ f ξ = i f ξ [ VII-39a ] ξ f ξ = i f ξ f ξ [ VII-39b ] where ξ= z L c L c = µ 0 ω χ E ω phase amplitue parts as [ ]. We next separate f ( ξ) f ( ξ) into R. Victor Jones, February, 000

13 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 70 f, ( ξ) = u, ξ [ ] [ VII-40 ] exp i φ, ( ξ) substitute into Equations [ VII-39a ] [ VII-39b ] -- iz. ξ u ξ + iu ( ξ) ξ φ ξ = i u ξ { [ φ ( ξ) ]} [ VII-4a ] exp i φ ξ ξ u ξ + iu ξ ξ φ ξ = i u ξ u ξ { [ φ ( ξ) ]} [ VII-4b ] exp i φ ξ Equating real imaginary parts of these equations, we fin ξ u ξ = u ( ξ) sin φ ( ξ) φ ( ξ) [ ] [ VII-4a ] u ( ξ) ξ φ ξ = u ( ξ) cos φ ( ξ) φ ( ξ) [ ] [ VII-4b ] ξ u ξ = u ( ξ) u ( ξ) sin φ ( ξ) φ ( ξ) ξ φ ξ = u ( ξ) cos φ ( ξ) φ ( ξ) [ ] [ VII-4c ] [ ] [ VII-4 ] Since the phase enters only in the combination ψ( ξ) = φ ( ξ) φ ( ξ) these four equations reuce to three -- iz. ξ u ξ = u ( ξ) sin ψ( ξ) [ VII-43a ] ξ u ξ = u ξ u ξ sin ψ ξ [ VII-43b ] R. Victor Jones, February, 000

14 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 7 ξ ψ ξ = u ( ξ) u ( ξ) ( ξ) Combining these three equations, we obtain u cos ψ ξ [ VII-43c ] tanψ( ξ) ξ ψ ξ u ( ξ) ξ u ξ u ξ ξ u ξ = 0 [ VII-44a ] which is, obiously, equialent to ξ ln u ξ [ u ( ξ) cos ψ( ξ) ] = 0 [ VII-44b ] or u ( ξ) u ξ cosψ( ξ) = const. [ VII-44c ] 0 as ξ 0 the "const' must be zero, hence, ψ( ξ) must be π for all Since u ξ ξ. Thus, the original four couple equations now reuce to two -- iz. ξ u ξ = u ( ξ) [ VII-45a ] ξ u ξ = u ξ u ξ [ VII-45b ] Combining these equations, we obtain u ( ξ) ξ u ξ + u ( ξ) ξ u ξ = ξ u [ ( ξ) + u ( ξ) ] = 0 [ VII-46a ] or R. Victor Jones, February, 000

15 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 7 u ( ξ) + u ( ξ) = [ VII-46b ] which is an assertion of the principle of energy conseration. Taking this equation together with Equation [ VII-45a ] we see that u ξ u ξ = ξ [ VII-47a ] or u ( ξ) = tanhξ [ VII-47b ] ln u ( ξ) = tanh( ξ) ξ u ( ξ) = cosh ξ [ VII-47c ] Finally, returning to the original ariables, we see that E ( z,ω ) = E( z,ω ) = i E( ω ) tanh( z L c ) [ VII-48a ] E ω cosh z L c [ VII-48b ] R. Victor Jones, February, 000

16 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 73 where L c = µ 0 ω χ E( ω ) R. Victor Jones, February, 000