High-resolution Interferometric Diagnostics for Ultrashort Pulses

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8. QUANTUM-PATH INTERFEROMETRY IN HIGH-HARMONIC GENERATIONture which could be systematically applied to the entire dataset. To do so, I included a quadraticterm in the power series expansion of the control-field dependent action:S (j ) (ω) ≈ S (j )D + ¯θ T ¯S (j )C + 1 2 ¯θ T (j ) ¯θ . (8.17)Here (j ) is a matrix which describes all combinations of quadratic self- and cross- couplingsbetween the control fields. I then obtained (j ) by fitting (8.17) to the exact quantum orbit analysis.In general, ¯θ T (j ) ¯θ describes a paraboloid, the contours of which are arbitrarily aligned tothe coordinate axes. The direction and magnitudes of the principal axes are given by diagonalization.Here, I not concerned with the details of the geometry, but only the magnitude of thecurvature. So, without loss of generality one may choose a co-ordinate system in which (j ) isdiagonal with largest eigenvalue ( (j ) ) 11 . This eigenvalue is equal to the largest quadratic coefficientover all possible directions in ¯θ space, and therefore forms a convenient measure of thecurvature of the action. This quantity is plotted for the long and short first-order trajectories inFig. 8.7, and for the long trajectory (the worst case) is around 1000. One obtains a feeling for thesignificance of this number by noting that the erroneous phase introduced by the nonlinearity isequal to 1/2( (j ) ) 11 θ1 2 . The control field amplitude at which the error equals one radian is therefore2/( (j ) ) 11 ≈ 0.045 for the monochromatic example used throughout thischapter.8.5.2 Choice of control-field amplitude scan rangeThe previous section showed that, away from cutoff, nonlinearity of action change is the dominantdistortion. The other factor that influences the width of the orbits in control-field-sensitivityspace is the intrinsic resolution in that space, determined by the range of control fields over whichD(ω; ¯θ ) is known. One may represent this mathematically via a window function ( ¯θ ) which isunity at the origin ¯θ = 0 and smoothly decays when approaching the edge of the sampled rangeof control field amplitudes. For the discussion which follows it is necessary to choose a particularform for the window function — here I use an isotropic Gaussian function ( ¯θ )=exp(− | ¯θ | 22w 2 ) (8.18)202
8.5 Limitation of the perturbative quantum path analysiswhere w = w FWHM /(2 2log2) parameterises the width of the window. Incorporating the windowfunction and the nonlinearity in the action into (8.11) gives(ω; ¯θ)= (j ) (ω) ( ¯θ ; (j ) )e iS(j )ω,D ei ¯θ T (j ) ¯S C . (8.19)jwhere I have incorporated the window function and the effect of the nonlinearity into a singlefunction ( ¯θ ;)=exp(− | ¯θ | 22w 2 + i 1 2 ¯θ T ¯θ ). (8.20)The Fourier transform is then˜(ω;ᾱ)=(2π) −L j (j ) (ω) ˜(ᾱ − ¯S (j )ω,C ;(j ) )iS(jω,C)e ω,D (8.21)where the Fourier transform of ( ¯θ ;) isi ˜L (ᾱ;)=det exp − 1 2ᾱT (w −2 I − i ) −1 ᾱ . (8.22)To find the spread of ˜ (ᾱ;) one may temporarily adopt a co-ordinate system in which is diagonalized.For simplicity I also assume that is real. Then˜ (ᾱ;)=⎡i Ldet exp ⎣− 1 2Ll =1α 2 l⎤w −2 + i ll⎦w −4 + 2 (8.23)llHere, ll are the eigenvalues of . The spread ∆α l of ˜ (ᾱ;) along α l is the square root of theinverse of the real part of the corresponding coefficient in (8.23)∆α l =w −4 + 2 llw −2 . (8.24)Minimization of (8.24) provides a natural choice of window function widthw = max −1/2lll(8.25)203
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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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2.4 Extending ultrashort metrology
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3 Phase reconstruction algorithm fo
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3.1 Motivationration of information
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3.2 Quantitative noise analysis for
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3.3 Nature of the input datato give
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3.4 Sampling and uniqueness of solu
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3.5 Main algorithmfrom all the shea
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3.5 Main algorithmsituation, Γ k ,
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3.5 Main algorithmfor j = 1,2,...,k
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3.6 Implications for the relative p
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3.7 Signal-to-noise ratio cost of a
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3.8 Signal-to-noise ratio benefit o
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3.9 Numerical comparison of single-
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3.10 Summary, critical evaluation,
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4 Experimental implementations of m
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4.1 Sequential acquisition of shear
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4.1 Sequential acquisition of shear
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4.2 Simultaneous acquisition of she
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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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Page 241 and 242: References[1] Rakov, V. & Uman, M.
Page 243 and 244: REFERENCES247 (1999). 11[63] Stiben
Page 245 and 246: REFERENCES[127] Gyuzalian, R., Sogo
Page 247 and 248: REFERENCES[189] Kornelis, W., Biege
Page 249 and 250: REFERENCES[254] Geindre, J. P., Aud
Page 251 and 252: REFERENCES[313] L’Huillier, A. &
Page 253 and 254: REFERENCESPhys. Rev. Lett. 91, 1630
Page 255 and 256: REFERENCESMarangos, J. In Conferenc
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High-resolution Interferometric Diagnostics for Ultrashort Pulses

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