High-resolution Interferometric Diagnostics for Ultrashort Pulses

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3. PHASE RECONSTRUCTION ALGORITHM FOR MULTIPLE SPECTRAL SHEARINGINTERFEROMETRYis in the complex amplitude itself — the sampling theorem givesE (ω)= E (ω n )sinc( ω − ω n). (3.35)ΩnHowever this is nonlinear in the sampled phase, preventing the use of matrix methods. One optionis to experiment with linear interpolants, which may perform acceptably under certain circumstances.A related concept has been studied and implemented in lateral shearing interferometry in twospatial dimensions. Here, the unknown phase φ(x,y ) is obtained using spatial shear measurementsof the form φ(x + X ,y + Y ) − φ(x,y ). At least two linearly independent shears are requiredfor complete wavefront characterization (up to the usual absolute phase ambiguity) [290]. Additionalshears can be used to improve the accuracy [291]. There are connections between theone-dimensional multiple shear case considered here and two-dimensional reconstructions: bothinvolve reconstructing the phase from an overdetermined set of wrapped phase differences. In themost common two-dimensional case, there are two orthogonal shears which define the samplingrate and the wavefront is assumed to not contain any vortices — singular points around which theclosed path integrals of the phase gradient are nonzero. In this situation, the unwrapping problemwhich occurred in this chapter does not arise since all phase differences are assumed to bewithin [−π,π]. However, if larger shears are used, or optical vortices are present, reconstructionalgorithms cannot assume the wrapped phase differences are the “true” values. Wavefront reconstructionin the presence of vortices, for example, require sophisticated algorithms [292] and are atopic of current research [293]. It is possible that the multiple-shear algorithm may shed light onsuch problems, or vice-versa.In the next chapter, I present the first experimental demonstrations of multiple SSI.92
4 Experimental implementations of multiple spectralshearing interferometryThis chapter presents the first experimental demonstrations of multiple spectral shearing interferometry(SSI). Two implementations of spectral phase interferometry for direct electric-field reconstruction(SPIDER) are used. For sequential acquisition of the shears, I used a spatially encoded arrangementSPIDER (SEA-SPIDER), the shear of which can be conveniently adjusted (without alteringany other parameters) using a delay stage. For simultaneous acquisition of multiple shears, I useda spatially encoded arrangement CAR-SPIDER (SEA-CAR-SPIDER), a recently developed version ofSPIDER that acquires a continuous range of shears mapped to the spatial dimension of an imagingspectrometer. In each case, I synthesized various test pulses and evaluated the precision of the reconstructionby examining the statistics over an ensemble of different acquisitions. The accuracywas tested either by comparison with a single shear reconstruction or by interfering the test pulsewith a simple, well-characterized reference.Tobias Witting provided assistance with the construction the apparatus used in this chapter,as well as with the data acquisition for section 4.1.4.1 Sequential acquisition of shearsAs a proof of principle, I compared the measurement of the spectral phase of a double pulse usingstandard concatenation and the multi-shear algorithm with two shears. Examination of thestatistics of the reconstructions obtained from an ensemble of acquisitions reveals a significantimprovement in the precision when using multiple shears. The work presented in this sectionformed part of the article “High precision self-referenced phase retrieval of complex pulses withmultiple-shearing spectral interferometry” [267].4.1.1 SEA-SPIDERI used a SEA-SPIDER [196], modified to use an external ancillary pulse rather than splitting off aportion of the test pulse. This variation, when applied to a traditional temporally encoded SPIDER,has previously been described as an X-SPIDER [81], modifed-SPIDER [285] or crosscorrelation SPI-93
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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
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2 BackgroundThis dissertation prese
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2.1 Metrology, ultrashort pulses, a
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2.1 Metrology, ultrashort pulses, a
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2.2 Formal introduction to ultrasho
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2.3 Introduction to ultrashort puls
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Page 53 and 54: 2.3 Introduction to ultrashort puls
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Page 75 and 76: 3 Phase reconstruction algorithm fo
Page 77 and 78: 3.1 Motivationration of information
Page 79 and 80: 3.2 Quantitative noise analysis for
Page 81 and 82: 3.3 Nature of the input datato give
Page 83 and 84: 3.4 Sampling and uniqueness of solu
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
Page 95 and 96: 3.8 Signal-to-noise ratio benefit o
Page 97 and 98: 3.9 Numerical comparison of single-
Page 103: 3.10 Summary, critical evaluation,
Page 107 and 108: 4.1 Sequential acquisition of shear
Page 109 and 110: 4.1 Sequential acquisition of shear
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Page 117 and 118: 5 Compact Space-time SPIDERIn this
Page 119 and 120: 5.2 Role of the measurement planetr
Page 121 and 122: 5.3 Analysis of ST-SPIDERof design
Page 123 and 124: 5.4 A common-path re-imaging ST-SPI
Page 131 and 132: 5.5 Experimental exampleRomero [295
Page 133 and 134: 5.6 Effect of miscalibrationy (pix)
Page 135: 5.7 Summary and outlooknow perform
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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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