2.1 Ultrafast solid-state lasers - ETH - the Keller Group

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124 2.1.7 Pulse characterization [Ref. p. 134 2.1.7.3.2 SPIDER Spectral Phase Interferometry for Direct Electric-field Reconstruction (SPIDER) measures the spectral interference of a pulse with a frequency-shifted replica [98Iac, 99Iac]: ∣ ∣ S (ω, τ) = ∣Ẽ (ω − ω 0)+Ẽ (ω − ω 0 − δω)e iωτ ∣∣ 2 , (2.1.90) where Ẽ (ω) is the Fourier-transformation of the electric field E(t). To access the spectral phase it is necessary to produce two time-delayed replicas by a predetermined delay τ of the pulse and with a spectral shear δω between their carrier frequencies. The spectral shear between the central frequencies of the two replicas is generated by upconversion of the two replicas with a strongly chirped pulse using sum-frequency generation in a nonlinear optical crystal. The strongly chirped pulse is generated by propagating another copy of the original pulse through a thick glass block. The spectral shear arises because the time delay between the replicas assures that each replica is upconverted with a different portion of the chirped pulse containing a different instantaneous frequency [99Gal]. The time delay τ between the replica is kept constant during the measurements and therefore only a 1D trace needs to be measured for the resulting SPIDER interferogram ∣ ∣ ∣∣ I SPIDER (ω) = ∣Ẽ(ω) 2 ∣ ∣∣ + Ẽ(ω + δω) ∣ 2 ∣ ∣ ∣∣ +2 ∣Ẽ(ω)Ẽ(ω + δω) cos [ϕ(ω + δω) − ϕ(ω)+ωτ] . (2.1.91) The phase information can be extracted from the cosine term by the following, purely algebraic, method: The Fourier transform to the time domain of the SPIDER interferogram consists of a peak at zero time and two side peaks located near τ and −τ. The two side peaks contain equivalent phase information and are equal. One of the peaks is isolated by applying a suitable filter function and an inverse Fourier transform back to the frequency domain yields a complex function z(ω), whose complex phase gives access to the pulse spectral phase: arg (z(ω)) = ϕ(ω + δω) − ϕ(ω) +ωτ. A separate measurement of the linear phase term ωτ by spectral interferometry of the short unsheared pulse replicas is subtracted from the previous expression to yield the spectral phase difference Δϕ(ω) =ϕ(ω + δω) − ϕ(ω). From an arbitrarily chosen starting frequency ω 0 one obtains the spectral phase ϕ(ω) at evenly spaced frequencies ω i = ω 0 + i · δω by the following concatenation procedure: ϕ(ω 0 )=ϕ 0 , ϕ(ω 1 )=Δϕ(ω 0 )+ϕ(ω 0 )=ϕ(ω 0 + δω) − ϕ(ω 0 )+ϕ(ω 0 )=ϕ(ω 0 +1· δω) , . ϕ(ω i+1 )=Δϕ(ω i )+ϕ(ω i ) . (2.1.92) The constant ϕ 0 remains undetermined but is only an offset to the spectral phase, which does not affect the temporal pulse shape and we may thus set it equal to zero. The spectral phase is written as: ϕ(ω i+1 )= i∑ Δϕ(ω k ) . (2.1.93) k=0 If δω is small relative to features in the spectral phase, Δϕ(ω) corresponds in first order to the first derivative of the spectral phase and we can approximate the spectral phase by: ϕ(ω) ≃ 1 ∫ ω Δϕ(ω ′ )dω ′ . (2.1.94) δω ω 0 The integral expression for the spectral phase has the advantage that all the measured sampling points of the interferogram can be used instead of only using a subset with sampling according Landolt-Börnstein New Series VIII/1B1
Ref. p. 134] 2.1 Ultrafast solid-state lasers 125 to the spectral shear as with the concatenation method. The phase information of the resulting interferogram allows the direct reconstruction of the spectral phase of the input pulse. Combined with the measured spectrum of the pulse the actual pulse can be calculated without any prior assumptions. Three design parameters determine the range of pulse durations that can be measured by a certain SPIDER apparatus: the delay τ, the spectral shear δω and the group-delay dispersion GDD up used to generate the strongly linearly chirped upconverter pulse. These three parameters are related as: δω = τ GDD up . (2.1.95) The delay τ, which determines the positions of the two side peaks of the Fourier transform of the interferogram, is chosen in such a way as to assure that the side peaks are well-separated from the center peak. On the other hand, the fringe spacing of the interferogram is proportional to 2π/τ and thus τ must be sufficiently small such that the spectrometer is able to fully resolve the fringes. The stretching factor GDD up is then chosen such that the spectral shear δω, which determines the sampling interval of the reconstructed spectral phase, is small enough to assure correct reconstruction of the electric field in the time domain according to the Whittaker–Shannon sampling theorem [00Dor]. The constrained relationship for τ and δω expressed by (2.1.95) means that with a particular SPIDER setup, only pulses with a limited range of pulse durations can be measured. A set-up for the sub-10-femtosecond regime in described in [99Gal]. 2.1.7.3.3 Comparison between FROG and SPIDER techniques The highest acquisition rates for single-shot pulse characterization reported so far were 1 kHz using SPIDER [03Kor] and 30 Hz using FROG [03Gar]. FROG is not well suited for high acquisition rates for two reasons: FROG requires the measurement of a 2D trace which makes the acquisition itself inherently slow and moreover the FROG algorithm uses an iterative process which requires a minimum number of steps for convergence depending on the complexity of the measured pulse. SPI- DER however only requires the acquisition of two 1D spectra: the so-called SPIDER interferogram and the optical spectrum of the pulse. Furthermore, the SPIDER algorithm is deterministic since it is based on two Fourier transforms with intermittent spectral filtering to reconstruct the spectral phase of the pulse. An additional Fourier transform is then required to calculate the electric field in the time domain and therefore the pulse reconstruction is fast. SPIDER is thus well suited for high acquisition rates and real-time operation. SPIDER has been shown to achieve high accuracy, i.e. the reconstructed electric field matches well with the physical field of the pulse [02Dor1], and high precision [02Dor2], implying a small spread between several reconstructions of the field obtained from the same data. In particular the SPIDER technique is reliable in the presence of limited experimental noise [00And, 04Jen]. SPIDER offers more bandwidth than any other technique and we even can measure pulses in the single-cycle regime [03Sch]. In addition, SPIDER still gives valid results even if the beam profile is not spatially coherent anymore because we can spatially resolve these measurements [01Gal]. A direct comparison between FROG and SPIDER techniques in the sub-10-femtosecond regime is given in [00Gal2]. Only fully characterized pulses in phase and amplitude will provide reliable information about pulse shape and pulse duration – this becomes even more important for pulses with very broad and complex spectra. In such a situation any other technique, such as fitting attempts to IAC traces, is very erroneous and generally underestimates the pulse duration. Landolt-Börnstein New Series VIII/1B1
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Page 103 and 104: References for 2.1 135 74New 74Sha
Page 105 and 106: References for 2.1 137 87Fan 87For
Page 107 and 108: References for 2.1 139 90Wu Wu, M.C
Page 109 and 110: References for 2.1 141 92Lem Lemoff
Page 111 and 112: References for 2.1 143 94Lin2 94Liu
Page 113 and 114: References for 2.1 145 95Kno Knox,
Page 115 and 116: References for 2.1 147 96Xu1 96Xu2
Page 117 and 118: References for 2.1 149 98Gab Gabel,
Page 119 and 120: References for 2.1 151 99Kub Kubece
Page 121 and 122: References for 2.1 153 00Car Carrig
Page 123 and 124: References for 2.1 155 01Hol Holzwa
Page 125 and 126: References for 2.1 157 02Kra1 02Kra
Page 127 and 128: References for 2.1 159 03Kor 03Kow
Page 129 and 130: References for 2.1 161 04Gra Grange
Page 131 and 132: References for 2.1 163 04Zha2 05Agn
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2.1 Ultrafast solid-state lasers - ETH - the Keller Group

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