High-resolution Interferometric Diagnostics for Ultrashort Pulses

2.3 Introduction to ultrashort pulse metrologyis the Wigner function of pulse E (t ), then application of a quadratic spectral phase modulationshears the Wigner function along the time axis yielding W (t −φω,ω), whilst a quadratic temporalphase modulation shears the Wigner function along the frequency axis yielding W (t ,ω+ψt ). Anyarea and orientation preserving linear transformation in chronocyclic space can be composed of aseries of such operations 1 . Temporal intensity is the projection of the Wigner distribution onto thetime axis I (t )=(2π) −1 ∞−∞onto the frequency axis Ĩ (ω)=(2π) −1 ∞W (t ,ω) dω; the spectrum is the projection of the Wigner distribution−∞W (t ,ω) dt .A particularly important case is when the shears are combined to produce a series of rotations.For each rotation, a projection of the Wigner function is obtained. The reconstruction of anarbitrary n-dimensional function from a series of projections onto lower dimensional datasets iscalled tomography [218] and is of fundamental importance in medical imaging [219]. When thefunction in question is a Wigner distribution, the procedure is chronocyclic tomography [220]. Thishas not been experimentally demonstrated because of the difficulty in performing accurate andadjustable spectral and temporal phase modulations. Nonetheless, it would have an advantageover other methods because it directly measures the Wigner distribution, which is the lowestorderdescription of shot-to-shot fluctuations in a partially coherent pulse train. Other methodsassume a coherent pulse train with no shot-to-shot fluctuation.The assumption of a coherent pulse train provides the additional constraint that the unknownis a Wigner function of a single pulse. This enables reconstruction of the Wigner function froma limited set of projections [221, 222]. Such simplified chronocyclic tomography [223] has beendemonstrated by performing two small rotations with an electro-optic phase modulator and measuringthe resulting spectra.2.3.7.4 Limits to temporal resolutionThe resolution of a spatial imaging system is determined by the focal length f and the diameter Dof its lens. Together, these form the numerical aperture NA = D/f which determines the maximum1This result comes from group theory: the matrix representation of area- and orientation-preserving linear transformationsis special linear group SL 2 () of all 2×2 real matrices with determinant one. Shear matrices form a generatingset of this group — all elements of SL 2 () can be written as a product of shear matrices [217].51