High-resolution Interferometric Diagnostics for Ultrashort Pulses

Definitions and symbolsAll Fourier transforms in this dissertation are unitary and use angular frequency. Thesign of the transform is usually determined by the physical context: a time-to-frequencytransform has exponential factor e i ωt , whilst a space-to-wavenumber transform hasexponential factor e −ixk x . In other cases, the sign is explicitly stated. When two quantitiesare related by an integral transform, such as the Fourier transform or the Hankeltransform, they are distinguished by their argument e.g. E (t ) vs E (ω). Specifically, theFourier transforms of the analytic electric field areE (ω)= 1 2π ∞−∞E (k x )= 1 2π ∞−∞E (t )e i ωt dt E(t )= 1 2π ∞−∞E (x)e −ik x x dx E(x)= 1 2π ∞−∞E (ω)e −itω dωE (k x )e ixk xdk x .The zeroth-order Hankel transforms are ∞ ∞E (k T )= E (r )J 0 (k T r )r dr E(r )= E (k T )J 0 (k T r )k T dk T .00Several notational conventions are adopted throughout this dissertation. Boldfacesymbols denote vectorial quantities. Where both an italic roman and calligraphic versionof a symbol exist they refer to the analytic (complex-valued) and physical (realvalued)signals respectively. The following table illustrates these two conventions:RealAnalyticScalar EVector E