Chapter 10Pulse CharacterizationCharacterization of ultrashort laser pulses with pulse widths greater than20 ps can be directly performed electronically using high speed photo detec-tors and sampling scopes. Photo detectors with bandwidth of 100 GHz areavailable. For shorter pulses usually some type of autocorrelation or cross-correlation in the optical domain using nonlinear optical effects has to beperformed, i.e. the pulse itself has to be used to measure its width, becausetherearenoothercontrollableeventsavailableonsuchshorttimescales.10.1Intensity AutocorrelationPulse duration measurements using second-harmonic intensity autocorrela-tion is a standard method for pulse characterisation. Figure 10.1 shows thesetupforabackground freeintensity autocorrelation.Theinputpulseis splitin two, and one of the pulses is delayed by T. The two pulses are focussedinto a nonliner optical crystal in a non-colinear fashion. The nonlinear opti-cal crystal is designed for efficient second harmonic generation over the fullbandwidth of the pulse, i.e. it has a large second order nonlinear opticalsuszeptibility and is phase matched for the specific wavelength range.Wedo not consider the z-dependence of the electric field and phase-matchingeffects. To simplify notation, we onit normalization factors. The inducednonlinear polarization is expressed as a convolution of two interfering electric-fields Ei(t), E2(t) with the nonlinear response function of the medium, the333
Chapter 10 Pulse Characterization Characterization of ultrashort laser pulses with pulse widths greater than 20 ps can be directly performed electronically using high speed photo detectors and sampling scopes. Photo detectors with bandwidth of 100 GHz are available. For shorter pulses usually some type of autocorrelation or crosscorrelation in the optical domain using nonlinear optical effects has to be performed, i.e. the pulse itself has to be used to measure its width, because there are no other controllable events available on such short time scales. 10.1 Intensity Autocorrelation Pulse duration measurements using second-harmonic intensity autocorrelation is a standard method for pulse characterisation. Figure 10.1 shows the setup for a background free intensity autocorrelation. The input pulse is split in two, and one of the pulses is delayed by τ . The two pulses are focussed into a nonliner optical crystal in a non-colinear fashion. The nonlinear optical crystal is designed for efficient second harmonic generation over the full bandwidth of the pulse, i.e. it has a large second order nonlinear optical suszeptibility and is phase matched for the specific wavelength range. We do not consider the z—dependence of the electric field and phase—matching effects. To simplify notation, we omit normalization factors. The induced nonlinear polarization is expressed as a convolution of two interfering electric— fields E1(t), E2(t) with the nonlinear response function of the medium, the 333
334CHAPTER10.PULSECHARACTERIZATIONsecond order nonlinear susceptibility x(2).p(2)(t) αx(2)(t - ti,t - t2) ·E(ti) · E2(t2)dtidt2Imageremoveddueto copyrightrestrictions.Please see:Keller,U.,Uitrafast LaserPhysics,Institute of Quantum Electronics,Swiss Federal Institute ofTechnologyETHHonggerberg—HPT,CH-8093Zurich,Switzerland.Figure1o.l:Setupforabackgroundfreeintensityautocorrelation.Toavoiddispersionandpulsedistortionsintheautocorrelatorreflectiveopticscanbeand a thin crystal has to be used for measureing very short, typically sub-100fs pulses.We assume the material response is instantaneous and replace x(2)(t -ti,t - t2) by a Dirac delta-function x(2) . 8(t - ti) - o(t - t2) which leads top(2)(t) α Ei(t) · E2(t)(10.1)Due to momentum conservation, see Figure 10.1, we mayseparate the productE(t)·E(t-T) geometrically and supress a possible background coming fromsimple SHG of the individual pulses alone.The signal is zero if the pulsesdon't overlap.p(2)(t) α E(t) · E(t - T).(10.2)
334 CHAPTER 10. PULSE CHARACTERIZATION second order nonlinear susceptibility χ(2). P(2)(t) ∝ ZZ ∞ −∞ χ(2)(t − t1, t − t2) · E1(t1) · E2(t2)dt1dt2 Figure 10.1: Setup for a background free intensity autocorrelation. To avoid dispersion and pulse distortions in the autocorrelator reflective optics can be and a thin crystal has to be used for measureing very short, typically sub-100 fs pulses. We assume the material response is instantaneous and replace χ(2)(t − t1, t − t2) by a Dirac delta—function χ(2) · δ(t − t1) · δ(t − t2) which leads to P(2)(t) ∝ E1(t) · E2(t) (10.1) Due to momentum conservation, see Figure 10.1, we mayseparate the product E(t)· E(t−τ ) geometrically and supress a possible background coming from simple SHG of the individual pulses alone. The signal is zero if the pulses don’t overlap. P(2)(t) ∝ E(t) · E(t − τ ). (10.2) 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:
10.1.INTENSITYAUTOCORRELATION335Imageremoved due to copyright restrictions.Pleasesee:Keller,U.,UltrafastLaserPhysics,InstituteofQuantumElectronics,SwissFederal InstituteofTechnologyETHHonggerberg—HPT,CH-8093Zurich,Switzerland.Table 10.1: Pulse shapes and its deconvolution factorsrelatingFWHM,Tp,of thepulse toFWHM,TA,of theintensity autocorrelationfunctionThe electric field of the second harmonic radiation is directly proportional tothe polarization,assuming a nondepleted fundamental radiation and the useof thin crystals.Due tomomentum conservation, see Figure 10.1, we findA(t)A(t - T)[ dt(10.3)IAc(t) αX?I(t)I(t - T) dt,(10.4)αα
10.1. INTENSITY AUTOCORRELATION 335 Table 10.1: Pulse shapes and its deconvolution factors relating FWHM, τ p, of the pulse to FWHM, τ A, of the intensity autocorrelationfunction. The electric field of the second harmonic radiation is directly proportional to the polarization, assuming a nondepleted fundamental radiation and the use of thin crystals. Due to momentum conservation, see Figure 10.1, we find IAC(τ ) ∝ Z ∞ −∞ ¯ ¯ ¯ A(t)A(t − τ ) ¯ ¯ ¯ 2 dt . (10.3) ∝ Z ∞ −∞ I(t)I(t − τ ) dt, (10.4) 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:
336CHAPTER10.PULSECHARACTERIZATIONwith the complex envelope A(t) and intensity I(t) =[A(t)? of the input pulse.Thephotodetectorintegratesbecauseitsresponseis usually muchslowerthan the pulsewidth. Note, that the intenisty autocorrelation is symmetricbyconstruction(10.5)IAc(T) = IAc(-T).It is obvious from Eq.(10.3) that the intensity autocorrelation does not con-tain full information about the electric field of the pulse, since the phase ofthe pulse in the time domain is completely lost. However, if the pulse shapeis known the pulse width can be extracted by deconvolution of the correla-tion function. Table 10.1 gives the deconvolution factors for some often usedpulse shapes.10.2Interferometric Autocorrelation (IAC)A pulse characterization method, that also reveals the phase of the pulseis the interferometric autocorrelation introduced by J. C.Diels [2], (Figure10.2 a). The input beam is again split into two and one of them is delayed.However,now thetwo pulsesare sent colinearly into the nonlinear crystal.Only the SHG component is detected after the filter.ImageremovedduetocopyrightrestrictionsPlease see:KellerUUltrafastarPhyicsInstitutefQuantumlectonicswissFedralInstitutefchnloETH Honggerberg—HPT, CH-8093 Zurich, Switzerland.Figure 10.2:(a) Setup for an interferometric autocorrelation.(b)Delaystage, so that both beams are reflected from the same air/medium interfaceimposing the same phase shifts on both pulses
336 CHAPTER 10. PULSE CHARACTERIZATION with the complex envelope A(t) and intensity I(t) = |A(t)| 2 of the input pulse. The photo detector integrates because its response is usually much slower than the pulsewidth. Note, that the intenisty autocorrelation is symmetric by construction IAC(τ ) = IAC(−τ ). (10.5) It is obvious from Eq.(10.3) that the intensity autocorrelation does not contain full information about the electric field of the pulse, since the phase of the pulse in the time domain is completely lost. However, if the pulse shape is known the pulse width can be extracted by deconvolution of the correlation function. Table 10.1 gives the deconvolution factors for some often used pulse shapes. 10.2 Interferometric Autocorrelation (IAC) A pulse characterization method, that also reveals the phase of the pulse is the interferometric autocorrelation introduced by J. C. Diels [2], (Figure 10.2 a). The input beam is again split into two and one of them is delayed. However, now the two pulses are sent colinearly into the nonlinear crystal. Only the SHG component is detected after the filter. Figure 10.2: (a) Setup for an interferometric autocorrelation. (b) Delay stage, so that both beams are reflected from the same air/medium interface imposing the same phase shifts on both pulses. 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:
10.2.INTERFEROMETRICAUTOCORRELATION (IAC)337The total field E(t, +) after the Michelson-Interferometer is given by thetwo identical pulses delayed by T with respect to each other(10.6)E(t,T) = E(t+T) +E(t)= A(t+T)ejwc(+T)ej中cE + A(t)ejwtej冲cE.(10.7)A(t) is the complex amplitude, the term eiwot describes the oscillation withthe carrier frequency wo and Φce is the carrier-envelope phase. Eq. (10.1)writesp(2)(t, T) α (A(t + T)ejwe(t+T)ej@cE + A(t)ejwstej@cE)2(10.8)This is only idealy the case if the paths for both beams are identical. Iffor example dielectric or metal beamsplitters are used, there are differentreflections involved in the Michelson-Interferometer shown in Fig. 10.2 (a)leading to a differential phase shift between the two pulses. This can beavoided by an exactly symmetric delay stage as shown in Fig. 10.1 (b).Again, the radiated second harmonic electric field is proportional to thepolarizationE(t, T) α (A(t+ T)ejwe(+T)ejocE + A(t)ejwe(t)ejocE)?(10.9)The photo-detector (or photomultiplier) integrates over the envelope of eachindividual pulse(A(t+T)eje(+T)+ A(t)ewt)2)I()αdA(t + T)ej2we(++r)X+2A(t+ T)A(t)ejwe(++T)ejwet+A2(t)ej2wt/(10.10)Evaluation of the absolute square leads to the following expressionI() αA(t + T))4 + 4|A(t + T)A(t)? + /A(t)4+2A(t + T)|A(t)PA*(t)ejweT + c.c.+2A(t)IA(t + T)PA*(t + T)e-jueT + cC.c.+A(t + T)(A*(t)*ej2weT + c.c.dt .(10.11)
10.2. INTERFEROMETRIC AUTOCORRELATION (IAC) 337 The total field E(t, τ ) after the Michelson-Interferometer is given by the two identical pulses delayed by τ with respect to each other E(t, τ ) = E(t + τ ) + E(t) (10.6) = A(t + τ )ejω⊂(t+τ) ejφCE + A(t)ejωct ejφCE . (10.7) A(t) is the complex amplitude, the term eiω0t describes the oscillation with the carrier frequency ω0 and φCE is the carrier-envelope phase. Eq. (10.1) writes P(2)(t, τ ) ∝ ¡ A(t + τ )ejωc(t+τ) ejφCE + A(t)ejωct ejφCE ¢2 (10.8) This is only idealy the case if the paths for both beams are identical. If for example dielectric or metal beamsplitters are used, there are different reflections involved in the Michelson-Interferometer shown in Fig. 10.2 (a) leading to a differential phase shift between the two pulses. This can be avoided by an exactly symmetric delay stage as shown in Fig. 10.1 (b). Again, the radiated second harmonic electric field is proportional to the polarization E(t, τ ) ∝ ¡ A(t + τ )ejωc(t+τ) ejφCE + A(t)ejωc(t) ejφCE ¢2 . (10.9) The photo—detector (or photomultiplier) integrates over the envelope of each individual pulse I(τ ) ∝ Z ∞ −∞ ¯ ¯ ¯ ¡ A(t + τ )ejωc(t+τ) + A(t)ejωct ¢2 ¯ ¯ ¯ 2 dt . ∝ Z ∞ −∞ ¯ ¯ ¯ A2 (t + τ )ej2ωc(t+τ) +2A(t + τ )A(t)ejωc(t+τ) ejωct +A2 (t)ej2ωct ¯ ¯ ¯ 2 . (10.10) Evaluation of the absolute square leads to the following expression I(τ ) ∝ Z ∞ −∞ h |A(t + τ )| 4 + 4|A(t + τ )| 2 |A(t)| 2 + |A(t)| 4 +2A(t + τ )|A(t)| 2 A∗ (t)ejωcτ + c.c. +2A(t)|A(t + τ )| 2 A∗ (t + τ )e−jωcτ + c.c. +A2 (t + τ )(A∗ (t))2 ej2ωcτ + c.c. i dt . (10.11)
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