High-resolution Interferometric Diagnostics for Ultrashort Pulses

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2. BACKGROUNDbe used, which projects the beam onto an approximation of a spatial Dirac delta function. Alternatively,a single-mode fibre projects the beam onto the fundamental mode of the fibre. Fromthese well-defined modes, diffraction produces a well-characterised spreading of the beam to anydesired spatial extent. A specific implementation of this is point diffraction interferometry [232].2.4.1.3 Spatial shearing interferometersSpatial shearing interferometry involves applying a small geometric transform to a beam, and theninterfering the transformed beam with the original. The measured phase is a finite difference,which is commonly treated as a derivative along the shear direction. Concatenation or integration,appropriately generalised to two dimensions, then recovers the phase.Lateral shearing interferometry (LSI) [233–237] is the most common form of spatial shearing interferometry,and is an exact analogue of spectral shearing interferometry. The beam is interferedwith a displaced version of itself, yielding the finite difference φ(x +X ,y )−φ(x,y ) ≈ X ∂φ , where for∂ xsimplicity I take the shear to be along the x-axis. Integration along the x-axis gives φ(x,y )+f (y ),where f (y ) is an unknown function resulting from the constant of integration of each line. Touniquely define the phase, two linearly independent shears are therefore needed [238].Methods of implementing the shear include a Mach-Zehnder [233, 238] or Michelson interferometer[235] with displaced output beams, a parallel or slightly wedged plate [234], or a diffractiongrating [239]. Implementations have been proposed using atom interferometry [240].Radial shearing interferometry. In LSI, the interferogram does not cover the whole area of thewavefront. Furthermore two shears are required for a complete characterisation. These limitationsled to the development of radial shearing interferometers, in which the beams being interferedare replicas of the original with a slightly different magnification. In a circular co-ordinatesystem (r,θ ), phase difference is therefore φ(mr,θ ) − φ(r,φ) ≈ (m − 1)r ∂φ . Excluding difficulties∂ rat the origin, where the shear drops to zero, the two-dimensional phase may be uniquely reconstructedby integrating along radial lines, since each share the origin at a common reference point.Implementations involve placing a telescope in either a Mach-Zehnder or Sagnac interferometer[241–243].56
2.4 Extending ultrashort metrology to the space-time domainRotational shearing interferometry [244] can also be performed, yielding the azimuthal phasederivative. A continuously adjustable rotation is provided by a Dove prism.2.4.2 Spatially resolved temporal measurementsA limited form of spatio-temporal characterisation involves performing many one-dimensionaltemporal characterisations on different regions of the beam. One may simply scan a pinhole (withsignificant loss of sensitivity) in the transverse plane, performing temporal characterisations onthe spatially selected region. Alternatively, if the dataset for each temporal characterisation isone-dimensional, the other dimension may provide spatial resolution, enabling temporal characterisationsto be performed in parallel at each spatial point. Spectral shearing interferometry iswell suited for this context [58, 59, 196, 197].The limitation of all such methods is that the ambiguities of the temporal reconstruction becomeunknown functions of space. For example, in self-referenced methods the near-universalabsolute phase ambiguity means that the recovered phase φ(x,y ,ω) contains an unknown additiveterm f (x ,y ), providing no information on spatial wavefronts and preventing numerical propagationof the pulse. The arrival time ambiguity prevents the diagnosis of pulse-front tilt. Specificexamples of this approach include spatially resolved crosscorrelations for the characterisationof multi-dimensional pulse shapers [72, 245], and SPIDER measurements revealing positiondependentduration and temporal profiles of few-cycle pulses obtained through filamentation[66].Another approach, free from the aforementioned ambiguities, is spatially encoded arrangementTADPOLE (SEA-TADPOLE) [246–248]. The temporal measurement is externally referencedFTSI, which is free of ambiguities provided the reference contains every frequency of the unknownpulse. SEA-TADPOLE therefore represents a complete but nonself-referenced spatio-temporalcharacterisation. A schematic is shown in Fig. 2.18. The reference pulse, which must also be characterisedby a separate device, is coupled into a single mode fibre. The unknown pulse is also coupledinto a separate single-mode fibre. However, the mode area of the fibre is much smaller thanthe beam size. The fibre therefore selects a point in the beam, which can be scanned in the trans-57
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High-resolution InterferometricDiag
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AcknowledgementsMy supervisor Ian W
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6 High-harmonic generation 1256.1 D
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List of acronymsADK Ammosov, Delone
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Definitions and symbolsAll Fourier
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1. INTRODUCTIONprofile in space as
Page 17 and 18: 2 BackgroundThis dissertation prese
Page 19 and 20: 2.1 Metrology, ultrashort pulses, a
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Page 23 and 24: 2.2 Formal introduction to ultrasho
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Page 75 and 76: 3 Phase reconstruction algorithm fo
Page 77 and 78: 3.1 Motivationration of information
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Page 81 and 82: 3.3 Nature of the input datato give
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Page 85 and 86: 3.5 Main algorithmfrom all the shea
Page 87 and 88: 3.5 Main algorithmsituation, Γ k ,
Page 89 and 90: 3.5 Main algorithmfor j = 1,2,...,k
Page 91 and 92: 3.6 Implications for the relative p
Page 93 and 94: 3.7 Signal-to-noise ratio cost of a
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Page 97 and 98: 3.9 Numerical comparison of single-
Page 103 and 104: 3.10 Summary, critical evaluation,
Page 105 and 106: 4 Experimental implementations of m
Page 107 and 108: 4.1 Sequential acquisition of shear
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Page 111 and 112: 4.2 Simultaneous acquisition of she
Page 117 and 118: 5 Compact Space-time SPIDERIn this
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5.2 Role of the measurement planetr
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5.3 Analysis of ST-SPIDERof design
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5.4 A common-path re-imaging ST-SPI
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5.5 Experimental exampleRomero [295
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5.6 Effect of miscalibrationy (pix)
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5.7 Summary and outlooknow perform
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6. HIGH-HARMONIC GENERATIONcurrent
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6. HIGH-HARMONIC GENERATIONcomponen
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6. HIGH-HARMONIC GENERATION00−I p
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6. HIGH-HARMONIC GENERATION (t ) (E
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6. HIGH-HARMONIC GENERATIONstationa
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6. HIGH-HARMONIC GENERATION10080log
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6. HIGH-HARMONIC GENERATIONwhere re
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6. HIGH-HARMONIC GENERATION• Sadd
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6. HIGH-HARMONIC GENERATIONgiven in
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6. HIGH-HARMONIC GENERATIONderivati
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6. HIGH-HARMONIC GENERATIONPhase ma
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6. HIGH-HARMONIC GENERATION30(S31)2
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6. HIGH-HARMONIC GENERATIONwhere W
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6. HIGH-HARMONIC GENERATIONtheir re
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6. HIGH-HARMONIC GENERATIONsmall pi
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7. LATERAL SHEARING INTERFEROMETRY
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8. QUANTUM-PATH INTERFEROMETRY IN H
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9 Summary and outlookThis chapter s
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the vacuum chamber on the laser pul
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A. NOISE AND UNCERTAINTYdomain, so
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A. NOISE AND UNCERTAINTYUsing the i
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B. SIMULATION CODES FOR HIGH-HARMON
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References[1] Rakov, V. & Uman, M.
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REFERENCES247 (1999). 11[63] Stiben
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REFERENCES[127] Gyuzalian, R., Sogo
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REFERENCES[189] Kornelis, W., Biege
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REFERENCES[254] Geindre, J. P., Aud
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REFERENCES[313] L’Huillier, A. &
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REFERENCESPhys. Rev. Lett. 91, 1630
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REFERENCESMarangos, J. In Conferenc
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High-resolution Interferometric Diagnostics for Ultrashort Pulses

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