Laser Beam Propagation

5 Characterization of Time Domain Pulses 5.1 Basics.............................................. 141 Two-Dimensional Distribution Functions • Filter Theory • Square Law Detector 5.2 Single-Path Detection Schemes ................ 152 Energy- and Power-Meter • Spectrometer • nth-Order Intensity Correlation • Sonogram • Spectrogram: FROG • Spectrally and Temporally Resolved Frequency Conversion: STRUT 5.3 Two-Path Detection Schemes .................. 166 Linear Interferometry • nth-Order Interferometric Correlation • Spectral Interferometry • Spectral Shearing Interferometry: SPIDER Thomas Feurer 5.4 Summary ......................................... 172 University of Bern References................................................ 173 This chapter aims to present a structured overview of the numerous methods and techniques employed to characterize ultrashort laser pulses. The variety of techniques is quite substantial and the common goal of all these techniques is to determine the electric field of an ultrashort laser pulse as precisely as possible. This quest is not as easy as it sounds and it took quite a number of years before practical, robust, and accurate methods emerged. Henceforth, I assume that the coherent laser pulses propagate preferentially in one direction and that their electric field vector has only one transverse component, that is, the pulses are linearly polarized. Additionally, I assume that the relevant real-valued electric field component can be decomposed in a product separating the spatial from the temporal part, that is, E(x, t) = f(x)E(t). While the spatial intensity distribution ∝|f(x)|2 can often be obtained by means as simple as a charge-coupled device (CCD) camera, the temporal part is more difficult to measure. Before describing suitable techniques, I 139 © 2014 by Taylor & Francis Group, LLC 140 Laser Beam Propagation introduce the analytic form of the electric field + − E(t) = E (t) + E (t). (5.1) + Moreover, the analytic field with amplitude |E (t)| and phase ϕ(t) is split in a slowly varying envelope times an oscillating carrier frequency + + ϕ( ) 1 + ω E (t) =|E (t)| ei t = E (t) ei 0t. (5.2) 2 + For ultrashort pulses, both the carrier phase ω0t and the slowly varying envelop E (t) vary so rapidly with time that no device is able to measure their time dependence directly. While the carrier frequency ω0 can be measured by a spectrometer, the slowly varying envelope requires indirect methods. The first derivative of the phase ϕ(t) with respect to time is the instantaneous frequency ω(t) + dϕ(t) darg(E (t)) ω(t) = = ω0 + . (5.3) dt dt The spectral fields + + iφ(ω) 1 + E (ω) =|E (ω)| e = E (ω − ω0) (5.4) 2 are obtained through Fourier transformation. The spectral phase is often expanded in a Taylor series: ∞ φ 1 d m φ(ω) = (ω − ω0) . (5.5) m! dω ω m=0 0 The lowest-order phase derivative (m = 0) is called absolute phase, m = 1linear phase, m = 2 quadratic phase (linear chirp), m = 3 cubic phase (quadratic chirp), and so on. To illustrate the techniques, I present analytic results for a Gauss-shaped input pulse with a quadratic phase modulation, that is 2 + (ω − ω ) E (ω) = exp − 0 (1 + iC) Gauss ω2 (5.6) ω2 Downloaded by [ETH BIBLIOTHEK (Zurich)] at 05:30 10 July 2014 + 1 E (ω) = exp − (1 + iC) Gauss 2 ω2 with a spectral bandwidth ω, a carrier frequency ω0, and a quadratic phase (linear chirp) C. Note that the spectral bandwidth is unaffected by the phase modulation. The corresponding temporal electric fields are 2 2 + ω t E (t) ∝ exp − (1 − iC) + iω0t Gauss 4(1 + C2) (5.7) 2 2 + ω t E (t) ∝ exp − (1 − iC) . Gauss 4(1 + C2) © 2014 by Taylor & Francis Group, LLC Characterization of Time Domain Pulses 141 The analytic results are useful to verify numerical codes simulating signals for arbitrary pulses. Moreover, I present simulated results for an asymmetric double pulse, a pulse with a quadratic, a cubic, and a quartic spectral phase modulation, and a pulse that has undergone self-phase modulation. Such pulses are frequently encountered in the daily lab work and the simulations help to identify them. In the following, I classify the different methods to characterize laser pulses in the general framework of filter theory. This approach is very similar to that published by Rolf Gase [1], but is used in many other contexts as well. Each element within a measurement setup is represented by a transfer function and there are rules how to calculate the electric field after the element from that before the element. A sequence of such elements plus a detector then comprises a single measurement setup. I also consider interferometric-type measurement techniques, where the incoming pulse is split in two or more replicas each traveling through a different sequence of elements before being recombined and analyzed by the detector. Next to classifying, my second goal is to indicate what type of information (intensity, electric field amplitude, phase etc.) is to be expected from a specific measurement technique and to show generic experimental setups. 5.1 Basics Before starting, I briefly introduce the concept of joint time–frequency distribution functions. They are useful, first, to visualize rather intuitively the dynamic evolution of the electric field and, second, because some of the measurement techniques produce such joint time–frequency distributions. So far, we have described the electric field as a function either of time or of frequency alone. Joint time–frequency distributions aim to display both aspects simultaneously. However, the time–energy (frequency) uncertainty principle of waves prohibits that both aspects can be shown with infinite resolution. Still, we will see that even under this constraint, the graphic information gain is extremely valuable. 5.1.1 Two-Dimensional Distribution Functions Downloaded by [ETH BIBLIOTHEK (Zurich)] at 05:30 10 July 2014 Figure 5.1 shows one of the oldest known joint time–frequency distributions. As time goes on (x-axis), the frequency (y-axis) increases and so does the amplitude. What we see here is the acoustic analog of a linearly chirped pulse train. The joint time–frequency distribution functions introduced here are only a subset of those known—the Wigner distribution, because it is one of the most widely used, and the spectrogram and the Page distribution, because they are related to some of the measurement techniques. An excellent review on the subject was published by Leon Cohen [2]. A further important question to pose in the context of pulse characterization is the following. Suppose a measurement has produced a specific time–frequency distribution, say a spectrogram; is it then possible to extract all information on the electric field from it? © 2014 by Taylor & Francis Group, LLC 142 Laser Beam Propagation Amplitude Frequency Frequency Amplitude Time Amplitude Time FIGURE 5.1 An ancient time–frequency distribution still in use. The time is running from the left to the right, the frequency is increasing in discrete steps from the bottom to the top, and the amplitude is indicated above the tones. Clearly, the time–frequency distribution reveals much more information on the dynamic evolution than either the temporal intensity (bottom plot) or the spectrum (right plot). 5.1.1.1 General Approach Ideally, a generic joint time–frequency distribution function, C(t, ω), should relate to the intensity per time and per frequency interval and fulfill the following constraints: + dω C(t, ω) ∝|E (t)|2 (5.8a) + dtC(t, ω) ∝|E (ω)|2 (5.8b) dt dω C(t, ω) ∝ F. (5.8c) Integrating C(t, ω) with respect to frequency should yield a function proportional to the temporal intensity and integrating with respect to time should result in a function Downloaded by [ETH BIBLIOTHEK (Zurich)] at 05:30 10 July 2014 proportional to the spectrum. Finally, integration over both variables should yield the total fluence F. A very general form that fulfills all these requirements is 1 − θ − τω+ θ C(t, ω) = du dτ dθ e i t i i u (θ, τ) 4π 2 − τ + τ × E u − E u + , (5.9) 2 2 where (θ, τ) is referred to as the kernel. It is the kernel that defines a specific distribution and it is the kernel that determines the properties of the joint time–frequency © 2014 by Taylor & Francis Group, LLC Characterization of Time Domain Pulses 143 TABLE 5.1 Three Joint Time–Frequency Distributions with Their Corresponding Kernel and Analytic Forms Distribution Kernel (θ, τ) C(t, ω) 1 − + −iωτ Wigner 1 π dτ E (t − τ/2) E (t + τ/2) e 2 t 2 iθ|τ|/2 1 ∂ + −iωτ Page e π ∂ dτ E (τ) e 2 t −∞ ∗( − τ/ ) −iθu ( + τ/ ) 1 τ (τ − ) +(τ) −iωτ 2 Spectrogram duh u 2 e h u 2 2π d h t E e distribution. In addition, with the help of the kernel, it is possible to transform each distribution to others. The kernel may be a function of time, of frequency, or even of the signal itself. Table 5.1 lists the kernels of the three joint time–frequency distributions discussed here. The general approach to the joint time–frequency distributions allows to study their properties in a standardized way. As stated above, an essential question is whether it is possible to extract the electric field once a distribution has been measured. Unfortu- nately, the answer is only partially yes. In general, the electric field is obtained through the following procedure: + 1 M(θ, t) − θ / E (t) = dθ e i t 2, (5.10) 2πE−(0) (θ, t) with the characteristic function . θ + τω M(θ, τ) = dt dω C(t, ω) ei t i , (5.11) which is just the two-dimensional inverse Fourier transformation of the time–frequency distribution C(t, ω).

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