338CHAPTER10.PULSECHARACTERIZATIONThe carrier-envelope phase ce drops out since it is identical to both pulses.The interferometric autocorrelation function is composed of the followingterms(10.12)I(T) = Iback + Iint(T) + Iw(T) + I2w() .Background signal Iback:(IA(t +T)/4 +[A(t)) dt = 2I(t) dtIback(10.13)Intensity autocorrelation Iint(+):Iimt(T) = 4 /~ [A(t + T)PIA(t)P dt = 4 /I(t+T) I(t) dt(10.14)Coherence term oscillating with we: Iw(t):I() = 4 /~Re[(I(t) + I(t+ T) A*(t)A(t + T)ejur dt(10.15)Coherence term oscillating with 2we: I2w(t):/Re[A(t)(A(t + T)ej2ur| atIw(μ) = 2 /(10.16)E.(10.12)is often normalized relative to the background intensity Ibackresultingintheinterferometric autocorrelationtraceIint(), I(), I2w(T)IAc(T) = 1 + (10.17)IbackbackIbackEg:(10.17)is thefinal equation for the normalized interferometric autocorrelation. The term Iint() is the intensity autocorrelation, measured bynon-colinear second harmonic generation as discussed before. Therefore, theaveraged interferometric autocorrelation results in the intensity autocorrela-tion sitting on a background of 1.Fig. 10.3 shows a calculated and measured IAC for a sech-shaped pulse
338 CHAPTER 10. PULSE CHARACTERIZATION The carrier—envelope phase φCE drops out since it is identical to both pulses. The interferometric autocorrelation function is composed of the following terms I(τ ) = Iback + Iint(τ ) + Iω(τ ) + I2ω(τ ) . (10.12) Background signal Iback: Iback = Z ∞ −∞ ¡ |A(t + τ )| 4 + |A(t)| 4¢ dt = 2 Z ∞ −∞ I2 (t) dt (10.13) Intensity autocorrelation Iint(τ ): Iint(τ )=4 Z ∞ −∞ |A(t + τ )| 2 |A(t)| 2 dt = 4 Z ∞ −∞ I(t + τ ) · I(t) dt (10.14) Coherence term oscillating with ωc: Iω(τ ): Iω(τ )=4 Z ∞ −∞ Rehµ I(t) + I(t + τ ) ¶ A∗ (t)A(t + τ )ejωτ i dt (10.15) Coherence term oscillating with 2ωc: I2ω(τ ): Iω(τ )=2 Z ∞ −∞ Reh A2 (t)(A∗ (t + τ ))2 ej2ωτ i dt (10.16) Eq. (10.12) is often normalized relative to the background intensity Iback resulting in the interferometric autocorrelation trace IIAC(τ )=1+ Iint(τ ) Iback + Iω(τ ) Iback + I2ω(τ ) Iback . (10.17) Eq. (10.17) is the final equation for the normalized interferometric autocorrelation. The term Iint(τ ) is the intensity autocorrelation, measured by non—colinear second harmonic generation as discussed before. Therefore, the averaged interferometric autocorrelation results in the intensity autocorrelation sitting on a background of 1. Fig. 10.3 shows a calculated and measured IAC for a sech-shaped pulse
33910.2.INTERFEROMETRICAUTOCORRELATION(IAC)Imageremovedduetocopyrightrestrictions.Please see:Keller,U.,UitrafastLaserPhysics,InstituteofQuantum Electronics,SwissFederal InstituteofTechnologyETHHonggerberg—HPT,CH-8093Zurich,SwitzerlandFigure 10.3: Computed and measured interferometric autocorrelation tracesfor a 10 fs long sech-shaped pulse.Aswiththeintensityautorcorrelation,byconstructiontheinterferometricautocorrelationhastobealsosymmetric:(10.18)IIAc(T) = IIAc(-T)This is only true if the beam path between the two replicas in the setupare completely identical, i.e.there is not even a phase shift between thetwo pulses. A phase shift would lead to a shift in the fringe pattern, whichshows up very strongly in few-cycle long pulses. To avoid such a symmetrybreaking, one has to arrange the delay line as shown in Figure 10.2 b sothat each pulse travels through the same amount of substrate material andundergoes the same reflections
10.2. INTERFEROMETRIC AUTOCORRELATION (IAC) 339 Figure 10.3: Computed and measured interferometric autocorrelation traces for a 10 fs long sech-shaped pulse. As with the intensity autorcorrelation, by construction the interferometric autocorrelation has to be also symmetric: IIAC(τ ) = IIAC(−τ ) (10.18) This is only true if the beam path between the two replicas in the setup are completely identical, i.e. there is not even a phase shift between the two pulses. A phase shift would lead to a shift in the fringe pattern, which shows up very strongly in few-cycle long pulses. To avoid such a symmetry breaking, one has to arrange the delay line as shown in Figure 10.2 b so that each pulse travels through the same amount of substrate material and undergoes the same reflections. Keller, U., Ultrafast Laser Physics, Institute of Quantum Electronics, Swiss Federal Institute of Technology, ETH Hönggerberg—HPT, CH-8093 Zurich, Switzerland. Image removed due to copyright restrictions. Please see:
340CHAPTER 10.PULSE CHARACTERIZATIONAt T = O, all integrals are identical/ 1A(t)14dtIboack = 2 /Iimt(T = 0) = 2 / IA2(t)Pdt = 2 / IA(t)4dt = Iback(10.19)I.(T = 0) = 2 / IA(t)PA(t)A*(t)dt = 2 / IA(t)dt = IoackI2 (T = 0) = 2 / A(t)(A2(t)*dt = 2 / IA(t)dt = IoackThen, we obtain for the interferometric autocorrelation at zero time delayIiAc(↑)lmax = IIAc(O) = 8IAC( → ±) = 1(10.20)IAc(↑)min = 0This is the important 1:8 ratio between the wings and the pick of the IAC,which is a good guide for proper alignment of an interferometric autocorre-lator. For a chirped pulse the envelope is not any longer real. A chirp in thepulse results in nodes in the IAC. Figure 10.4 shows the IAC of a chirpedsech-pulse1+jBA(t) = ((sechfor different chirps
340 CHAPTER 10. PULSE CHARACTERIZATION At τ = 0, all integrals are identical Iback ≡ 2 Z |A(t)| 4 dt Iint(τ = 0) ≡ 2 Z |A2 (t)| 2 dt = 2 Z |A(t)| 4 dt = Iback Iω(τ = 0) ≡ 2 Z |A(t)| 2 A(t)A∗ (t)dt = 2 Z |A(t)| 4 dt = Iback I2ω(τ = 0) ≡ 2 Z A2 (t)(A2 (t) ∗ dt = 2 Z |A(t)| 4 dt = Iback (10.19) Then, we obtain for the interferometric autocorrelation at zero time delay IIAC(τ )|max = IIAC(0) = 8 IIAC(τ → ±∞)=1 IIAC(τ )|min = 0 (10.20) This is the important 1:8 ratio between the wings and the pick of the IAC, which is a good guide for proper alignment of an interferometric autocorrelator. For a chirped pulse the envelope is not any longer real. A chirp in the pulse results in nodes in the IAC. Figure 10.4 shows the IAC of a chirped sech-pulse A(t) = µ sech µ t τ p ¶¶(1+jβ) for different chirps
10.2.INTERFEROMETRICAUTOCORRELATION(IAC)341ImageremovedduetocopyrightrestrictionsPlease see:Keller,U.,UltrafastLaserPhysics,InstituteofQuantumElectronics,SwissFederal InstituteofTechnologyETH Honggerberg—HPT,CH-8093Zurich, Switzerland.Figure 10.4: Influence of increasing chirp on the IAC10.2.1Interferometric Autocorrelation of an UnchirpedSech-PulseEnvelope of an unchirped sech-pulseA(t) = sech(t/Tp)(10.21)Interferometric autocorrelation of a sech-pulse3 (() cosh (- sinh (-0.22IIAc(T) = 1 +[2 +cos(2wT)sinh ()3 (sinh ()- ()cOs(weT)sinh (e)
10.2. INTERFEROMETRIC AUTOCORRELATION (IAC) 341 Figure 10.4: Influence of increasing chirp on the IAC. 10.2.1 Interferometric Autocorrelation of an Unchirped Sech-Pulse Envelope of an unchirped sech-pulse A(t) = sech(t/τ p) (10.21) Interferometric autocorrelation of a sech-pulse IIAC(τ ) = 1+ {2 + cos (2ωcτ )} 3 ³³ τ τp ´ cosh ³ τ τp ´ − sinh ³ τ τp ´´ sinh3 ³ τ τp ´ (10.22) + 3 ³ sinh ³ 2τ τp ´ − ³ 2τ τp ´´ sinh3 ³ τ τp ´ cos(ωcτ ) Keller, U., Ultrafast Laser Physics, Institute of Quantum Electronics, Swiss Federal Institute of Technology, ETH Hönggerberg—HPT, CH-8093 Zurich, Switzerland. Image removed due to copyright restrictions. Please see:
342CHAPTER 10.PULSE CHARACTERIZATION10.2.2Interferometric Autocorrelation of a ChirpedGaussianPulseComplex envelope of a Gaussian pulseA(t) = exp -() (1+jp)(10.23)Interferometric autocorrelation of a Gaussian pulsellac(t) = 1 +[2 +e-%(#) cos(2w.r)e-()(10.24)+4e-() cos(() cos (w.T),10.2.3Second Order DispersionIt is fairly simple to compute in the Fourier domain what happens in thepresence ofdispersion.E(t) = A(t)ewet EE(w)(10.25)After propagation through a dispersive medium we obtain in the Fourierdomain.E(w) = E(w)e-i()andE(t) = A(t)ejwetFigure 10.5 shows the pulse amplitude before and after propagation througha medium with second order dispersion. The pulse broadens due to the dis-persion. If the dispersion is further increased the broadening increases andthe interferometric autocorrelation traces shown in Figure 10.5 develope acharacteristic pedestal due to the term Iint. The width of the interferomet-rically sensitive part remains the same and is more related to the coherencetime in the pulse, that is proportional to the inverse spectral width and doesnot change
342 CHAPTER 10. PULSE CHARACTERIZATION 10.2.2 Interferometric Autocorrelation of a Chirped Gaussian Pulse Complex envelope of a Gaussian pulse A(t) = exp ∙ −1 2 µ t tp ¶ (1 + jβ) ¸ . (10.23) Interferometric autocorrelation of a Gaussian pulse IIAC(τ ) = 1+ ½ 2 + e −β2 2 ³ τ τp ´2 cos(2ωcτ ) ¾ e −1 2 ³ τ τp ´2 (10.24) +4e − 3+β2 8 ³ τ τp ´2 cos à β 4 µ τ τ p ¶2 ! cos (ωcτ ). 10.2.3 Second Order Dispersion It is fairly simple to compute in the Fourier domain what happens in the presence of dispersion. E(t) = A(t)ejωct F −→ E˜(ω) (10.25) After propagation through a dispersive medium we obtain in the Fourier domain. E˜0 (ω) = E˜(ω)e−iΦ(ω) and E0 (t) = A0 (t)ejωct Figure 10.5 shows the pulse amplitude before and after propagation through a medium with second order dispersion. The pulse broadens due to the dispersion. If the dispersion is further increased the broadening increases and the interferometric autocorrelation traces shown in Figure 10.5 develope a characteristic pedestal due to the term Iint. The width of the interferometrically sensitive part remains the same and is more related to the coherence time in the pulse, that is proportional to the inverse spectral width and does not change