Конспект консультации Гаврикова по оптике

∆E =µ" ¨Ec2µ"@ 2 Ex= 2 Ëx@x2ccv=√ ,"µµ"f (x − vt) + g(x + vt)E(x, t) = A cos(!(t −x)) = A cos(!t − kx)cE(~r, t) = A(~r) cos(!t − ~k~r)~ · ei'(r) · ei!t .E(~r, t) = A(r)'(r) = ~k~rA(r) = const, '(r) = ~k~·rf (~r) = A(~r) · ei'(~r)E(~r, t) = f (~r) · e−i!t∆f + k 2 f = 0∆−E(r, t) = A(r)ei'−i!t'(r) = ~k · ~r = constE=A ikr −i!t·e ·r~k~r,rr,r2 .E 2 · r2E 2 r2 = const ⇒ E ∼1r1E∼√ ,rE1 = a1 (~r) · ei'1 (~r)−i!tE2 = a2 (~r) · ei'2 (~r)−i!tΘa1 sin '1 +a2 sin '2a1 cos '1 +a2 sin '2E3 = E1 + E2 = A · eiΘ(~r) · e−i!t .A2 = a21 +a22 +2a1 a2 cos('2 −'1 ), tan Θ =a1R0 ,a1R.R=r22R0 .a0r22(AP=)I = A2 + a20 + 2Aa0 cos(k)2R0AeikR0 e−i!ta1 kR0eR0 er2ik 2R0· e−i!t ,a1R0= a0 .pR0R02 + r2 ≈ R0 +|~k1 | = |~k2 | = k~E1 = a1 ei(k1 ~r−δ1 ) e−i!tδ2 = 0.'2 − '1 = (~k2 − ~k1 ) · ~r.k~2 ~k1|K| = 2k sin ↵/2,~ · ~r = '2 − '1 .K↵δ1 =~Kk1 k2√~ · ~r),I = I1 + I2 + 2 I1 I2 cos(K√2 I1 I2 cos(2k sin ↵2 · z)Imax = (a1 + a2 )2∆' = ⇡(2m+1).√~ · ~r I = I1 + I2 + 2 I1 I2 cos Kz = I1 + I2 +K∆' = 2k sin ↵2 · z∆' = 2⇡ · m1 ; Imin = (a1 − a2 )2k =l=V =Imax −IminImax +Imin2⇡λ2k sin ↵2 ·l = 2⇡,λλ≈ .2 sin ↵2↵Imax = (a1 + a2 )2 , Imin = (a1 − a2 )2T ∼ 10−15τ ∼ 10−10τ,∆ωτ,τ · c = ∆max∆max ,dI =2(Iω dω)(1 + cos k∆) = 2A2 (1 + cos k∆).I =cosω·∆c )´ωω+ ∆ω2− ∆ω2dI = 2I0 (1 +∆ω·∆2c∆ω·∆2csin·∆maxV =|∆ω·∆2c∆ω·∆2csin∆max =∆ω·τ = 2π.|2πc∆ω .2πc2πc∆ω = (2πc· ∆λ)λ2∆maxλ=λ∆λ=λ2∆λ∆max =mmax =↵l=λ↵.AD = y sin Ω2 ,AC = y · sin Ω2ΩdI ∼ (1 + cos 2⇡` · x) · dya1 a2b1∆a = AD;b1b2b2 ∆b = −AC.∆ = |∆a | + |∆b |∆∆ = 2y sin Ω2 ≈ y · Ω.∆ = λ,∆x = `,∆x)]·dy;.∆x = yΩ`λ´ b/2[1+cos( 2⇡` x − kyΩ)]dy = (1 +−b/2V =Ωb2= ⇡,bmax =λΩ.Imax −IminImax +ImindI = [1 + cos 2⇡` (x −∆xkΩb2kΩb2sin= |I = const ·· cossin kΩb2kΩb22⇡` x)· const|.2⇡λ·Ωmax = λb ..λb·L=λ⇢ = Ωmax · L.

=∆≤∆max =Ω≤λbλ2∆λψ,2∆E− c12 ∂∂tE2 = 0,E(P ) =f0 (x, y)´Sf0 (x, y) · 1r , ·eikr dσ · K(α)αK(α) = cos α · K0 K0 =f01iλλ,Dbλ ⌧ b, D bD,1P2 ;P =P ⇠ 1,P $ 1,P ⌧ 1,√λbD ,M0 , M1 , M2 ...M1 DP(ρ2 = 2axλbx = 2(a+b)p) ρm =M1 P = M0 P + λ2 ; M2 P = M1 P + λ2 . . .M1D. M0 P = b, SM0 = SM1 = a, M1 P = b + λ2 ; M1 D = ρ, M0 D = x.ρ2 = a2 − (a − x)2SM1 Dλ 222ρ = (b + 2 ) − (b + x) .λ, x ⌧ a, b,λ2 x 2 .qλabm a+bλ2mλbR = const = b,E(P ) =ikR´K(α) = constR =pK(α) · A0 eR dσ =´ρ20 ikbe .= [ eik( 2b ) · (2πρdρ)] · K(α)Ab´2ξ = ρ : E(P ) = eikξ/2b dξ · A.a ! 1 ) ρm =b 2 + ρ2 ⇡ b +ρ22b .a0 eiϕa0 = dξ,ϕ = kξ/2bK(α)  1,π.AdxdE = K(α) √· ei(kr−z) .Rλr=zr − z = (x2 + z 2 )0.5 − z ⇡x22zx2dE = K(α) · const · e−ik 2z · dx.I(x),I(x).I(x)x ! −1λ2λ2oN · π · A0Ip = A2p = π 2 A20 N 2 .A0P :k · (M F − (n · OQ + OF )) = 0ρ; OF = f ;ρ,ρ2 = 2f (n − 1)ξ + (n2 − 1)ξ 2pM F = n·OQ+OF(f + ξ)2 + ρ2 = nξ + f,OQ = ξ; QM =P =√λbD .D,P # 1,pλbD#pλbP ⌧ 1,ξ, η,(ξ, η)x, y,OP = R0 , M P = R, P = (x, y);xyE(P ) =f0 (ξ, η) ·eikRR· dξdη;K(α)f0 (ξ, η)R ⇡ R0 ,R=pz 2 + (x − ξ)2 + (y − η)2 ⇡ R0 −ξ 2 +η 22R0ξ 2 +η 22R0ξ 2 + η2  D2 .R ⇡ R0 −D ⌧ R0xξ+yηR0 ,˜K(α) ·xξ+yηR0+ξΘ,ξ∆.∆ = ξ sin ΘdE = E0 eik∆ dξ´ +D/2Ep = −D/2 E0 eikξ sin Θ dξ.ξsin xxI(Θ) = I0 [sinkΘD 2kΘD2 /( 2 )]: Ep =Θ ⇡ 0,1k sin2 Θ·(eik sin Θ2−e−I(Θ) = I0 [sinik sin Θ2)/(2i)·E0 .k sin ΘD k sin ΘD 2/ 2 ]2ΘΘ∗Θ∗ :kΘ∗ D2= π ) Θ∗ = λ/D.Θ = 0, 61λ/D.Θ∗ =λD12π´ +∞C(Ω) · eiΩx dx,−∞´ +∞C(Ω) = −∞ f (x) · e−iΩx dx.´ +∞1C(Ω) · eiΩx dxf (x) = 2π−∞f (x) =f (x)f (x)E(P ) =f (x)C(Ω)´xξeik R0 dξ .

. .η . . .const . . .Θ∆. ∆ = d sin Θ∆∆ϕ = k∆ =2πλ d sin Θ.ΘE1 = E0 sinα αE0E2 = E1 · e−i∆ϕ ,E3 = E2 · e−i∆ϕ .EP =⇒ EP = E1 · 1−e1−e−i∆ϕ = E1 ·−iN ∆ϕe−(N −1)i∆ϕ ).Θ.E. I = I1 · (sin N ∆ϕ2)2sin ∆ϕ2= I0 (sin k ΘD2)2k ΘD2α =·(sin N ∆ϕ2)2 .sin ∆ϕ2sin N ∆ϕ2sin ∆ϕ2Pk sin Θ·D2Ej = E1 ·(1+e−i∆ϕ +...+·e−i(N −1)∆ϕN2I1N2λ 2λd , d ...λNd(N − 1).λ1 > λD1 =D1 =md cos Θ .cos Θ ≈ 1,mλ = d sin Θ),D1 = md.dΘdλ .λ(m + 1) =∆λmax =(λ + ∆λ)mR=Rλδλ ,dΘ =dΘ.dΘdλδλ· δλ.dΘdΘ =λNd ,λNd=md δλλm.⇒ R = m·N.dΘdλ=D=md,dΘ =md δλ.n(λ)l2 −l1h·dndλ .R = (l2 − l1 ) dndλ ,D =A0t,∼ 0, 95r.Θ.A0 t,A0 tr,A0 t 2 ,AP .AP .A0 t2 r4 eik·2∆A0 t2 ,A0 t2 r2 · eik∆ ,∆:∆ = 2L · cos ΘPAP = j Aj =∗I = AP · AP==Q =Q =Q=W∆Wω∆ωW∆WI0 t 21+r 2 −2r 2 cos k∆A0 t 2;1−r 2 eik∆2L cos Θ = ∆ = mλ· 2πω· 2π =( λc )2πL(1−ρ)λW (1 −= R.λρ) L,ρ = r2λ,R.λ| = RQ = | ∆λLcW (1 − ρ).↵~ki~k~r= a0 e · e−i!t ,zA:~~E = ae−i(!t−kr) =~a0 eik~r = A~k~r,A = a0 ei(k sin ↵·x+k cos ↵·z) .A0 = a0 eik sin ↵·x .z = 0.a0 eiΩx .´ +∞12⇡−∞C(Ω)·eiΩx ·dΩ´ +∞C(Ω) = −∞ f (x) · e−iΩx dx.

C(Ω) · eiΩxΩf (x)Ω = k sin ↵ ) A0 =f (x) =z = 0 : C(Ω)eiΩx .f (x) =Pf (x)an eiΩn xa0sin ↵n =z = 0,Ωnk ,z 6= 0.z 6= 0.√ei(k sin ↵n ·x+pk2 −Ω2 ·z)k 2 − Ω2 = k cos ↵n ,f (x, z 6= 0)Pz = 0,f (x, z 6= 0) =an ·Pk sin ↵n = Ωn ) f (x, z 6= 0) =an ei(Ωn x+k cos ↵n z) .(z = 0)z 6= 0f (x)⇠sin xx ,f1 =´f0 ·eikx2R0·⇠d⇠.xf (x) = A0 (1 + m cos Ωx),Ωcos(Ωx) =iΩxA0 m2e+−iΩxA0 m.2esin α1 =1 iΩx2 (e+ e−iΩx ) ) f (x) = A0 +e0 ) Ω0 = 0 ) α0 = 0;Ω2 = −Ω,sin α2 = − Ωk.Ωk;α1Ω1 = Ω,α1α2z.f (x) = A0 (1 + im cos Ωx).πm i(−Ωx+ πi(Ωx+ π2) + A2 ).ei 2 ) f (x) = A0 + A0 m0 2e2eiΩx−iΩx+ iA0 m.f (x) = A0 + iA0 m2e2eπ2,i =π2zz 6= 0.f (x, z 6= 0) =m i(−Ωx+(k2 −Ω2 )1/2 ·z)i(Ωx+(k2 −Ω2 )1/2 ·z)e+Ae.A0 ei(0·x+kz) + A0 m0 22z,llz = 0.∆(ϕ)pk 2 − Ω2 · z,zp z,∆(ϕ) = kz − k 2 − Ω2 · zkz.∆ϕ = 0z = 0,∆ϕ =π2,∆ϕ = π,∆ϕ = 23 π,∆ϕπ2.∆ϕ = ππ2,π2π2,R0 ,r,A0 eikR0 · eρ.r ≈ R0 ,ρ2ik 2R0r=a0 eikR0pρ2R02 + ρ2 ≈ R0 + 2R0.2.ρI = A20 + a2 + 2A0 a cos(k 2R).0A0 = a,ρ2n : k 2Rm0 = π · m ⇒ ρm =√mλR0 .2ρτ (x) ∼ 1+cos(k 2R)0A0,A0 ·τ (x),2A0 · τ (x) = 1 +ρ1 ik 2R02eρ2+ 12 e−ik 2R0R0Θ.S = A(x, y) · eiψ(x,y)S−S0 = a0 eikx sin ΘI = A2 + a20 + Aa0 eiψ e−ik sin Θ·x + Aa0 e−iψ eikx sin Θ .iψ −ik sin Θ·xAa0 e e,Aa0 e−iψ ikx sin ΘeΘ.Θ.A2 + a20 ,.1n≥ρ,ρ2 = mλR0 .2ρdρ = λR0 dmmmm−1n⇒ ρ =dmdρnλR022ρb,ψ,b=ρ,R0 ,b=λψ=λ2ρ R0ρ,λψ.d.Θd =Θλ2 sin Θ ,dρ2 = mR0 λρ2ρdρ = mR0 dλ.λ,mm=ρ2R0 λ ,dρ =1n, ⇒dλ =2ρnmR0λ + ∆λ,~E~rE~E.~E~ t.E~H~rEE1τ = E2τ ; H(1τ = H2τ .E cos '1 − Er cos '1 = Et cos '2n 1 E + n 1 Er = n 2 Et(E − Er = Et;n1 E cos '1 + n1 E2 cos '1 = n2 E cos '2~E;Et√Er ;~~ = √"EHn 1 E + n 1 E r = n 2 Et ?µ~H" = n; H1τ = H2τ ⇒ n1 E + n1 Er = n2 EtR|| = ( EEr )2 =R⊥ =sin2 (ϕ2 −ϕ1 )sin2 (ϕ2 +ϕ1 )tan2 (ϕ1 −ϕ2 )tan2 (ϕ1 +ϕ2 ) ;R|| = 0,'1 + '2 =R||tan ' =n2n1 ;π2⇒ R|| = 0;'1 + ' 2 ='1π2~E.~EI0 ,I02 .↵I1 =I02cos2 ↵~EE cos ↵,E.z~D~ e.Dd.n0 .ne ,z,~o ⊥ zD~eD~oD∆' = k(ne − no ) · d;∆' = π2 ,dnenoλ4;∆',λ2;ne > no ,2⇡?2⇡,> ∆maxD,DEE?D,λ4λ4;ExEyEx = E yλ2ExEy ;zz~E.xyt~E~Et + ∆t~E.~E,E y ExEx = Ex0 cos !t; Ey = Ey0 cos(!t + ')'=~EEyExE x = Ey ;⇡2 , Ex0 = Ey0 ,ExEyEyExxy.~ = "E~D0 1 0"xxDx@Dy A = @"yx"zxDz"xy"yy"zy10 1Ex"xz"yz A @Ey AEz"zz0"x" = @00pp"x =0"y0100A"z"x = " y = " z"x = "y 6= "zp" y = no"x 6= "y 6= "z" z = nezxy,↵z~H~ E,~ ~k, S~ S~D,~k~Sz,x,8~i(!t−~kr)>< E = E0 eD = D0>:H = H0~ :D~v=!k=cn)) n = no ;~ D~E,D2=!2c2 k 2 2=cos2 ↵); D2n2022sin ↵cos ↵n2e + n20 ;1n2=n = n(↵),Dk2"k0z;v2D?"?= D02 ;2D?n2e2D?n20~ D0D="o@00=~~ = − 1 @HrotEc @t~~ = 1 @DrotH~~ = !H[~k ⇥ E]c~ = −!D~[~k ⇥ H]c~ D~E,D2z,z) =e(1n2~D2;~ : ~k(~k E)~ −E~ · ~k 2 = − !22 · D;~Hc2!22~ · D)k~ ? ~k ) (D,~ ~k) = 0 ) (E~D= c2 · D ;~ ?DD02 ( sinn2 ↵ +(c @t;!2c2 k 2 2 ;0"o0cn0100A"e~ = "E,~D= const~ ·D~ =E~ kD~ k +E~ ?D~? =E~ D~E,D2=~ · D)~ =(E1n2v=cn;λ4,λ/4Ex , Ey ;λ4,ExExλ4,λ2,Ey ,Ey ,ExEymr̈ = −kr − β ṙ +−krf (r) = −kr + a1 r2 + a2 r3 + ...eEβ=0E⇡E= 106 − 108r2mr̈ = −kr + a1 r + eE ) r̈ + ω02 · r − ( am1 ) · r2 =rP = N · e · r = Np :a1N e2a2)P=P̈ + ω0 P − ( mNem E(t);mN e2+P1 (t),P1 (t) : P̈ 1 + ω02 P1 = N aem (P0 + P1 )2 ;P1 ⌧ P0 ) P̈1 + ω02 P1 =1· α2 E02 cos2 ωt = βE02 + βE02 cos 2ωt.P1 P¨1 + ω02 P1 = Naemem E(t).P = P0 (t)P1 (t)P0 (t)a2N em P 0E2E20P : P (t) = αE cos ωt + β ω20 + β ω2 −4ω2 · cos 2ωt,00E2β ω2002E0,E2αE cos ωt2ω,0β ω2 −4ω2 · cos 2ωt02ω2ω2ωEei(ωt−kr) ,2ω,P = p0 cos(2ωt−2k1 r),2ω,E2 = E02 cos(2ωt − k2 r),2ω,k2 ,k2 6= 2k1 .2ω,2ω2ω∆ϕ = (k2 − 2k1 ) · z.∆ϕ ⌧ π,zz∆ϕ ⌧ π,∆ϕ ⇠ π,z,z=πk2 −2k1z,β 6= 0,z =πk2 −2k1=πc12ω n(2ω)−n(ω) .no (2ω)z = 1,E1 .2ωω1no (2ω) = ne (ω),ne (ω),r3 .P̈ + ω02 P −a23N 2 e2 m P=N e2m E0cos ωtP = P0+P3P3P3 : P̈3 + ω02 P3 = N 2ae22 m α3 E03 cos ωt.α(1 + b1 E02 ) · E0 cos ωt + b2 E03 cos 3ωt.ωE,cos3α(1 + b1 E02 ) · E0 cos ωt.3ω : P̈3 + ω02 P3 =n0 ,n1 = n0 + n01 E02 ,n 1 > n0 ,E02 =λ202n0 n1 d2 ,dE0 ,n(!)v=cn(!)v =!(k);!k,v@kk(!) = k0 +( @!)|0 (!−!0 ).v=@!@k= v −k@v@kv<vv~mr̈ = −kr −β ṙ +eE,~E = E0 eik~r ·ei!t ,~r>vr~E0 eik~rȦ = 10−10⌧λ)λ ⇡ 500~E0 eik~rr = r0 ei!t,r=e/m!02 −! 2 +2i!γE.r̈ + 2γ ṙ + !02 r = Aei!t ;· E0 ei!t,rp~ = e~r;"E = (1 +4⇡N ·e2 /m)!02 −! 2 +2i!γ~ = E~ + 4⇡ P~ = "E.~D4⇡N ·e2 /m" = 1 + !2 −!2 +2i!γ ;· E," = n − iκ!!0 .κ!0 ?pN~r" = n,0pnP~ = N · p~,nκ!0 = 0 γ = 02" = 1 − 4⇡N!e2 /m ,4⇡N e2 /m = !p2!2" = 1 − !p2 ,!0!p ,γ.