Regularization of an autoconvolution problem in ultrashort laser pulse characterization D. Gerth a,b, B. Hofmann b, S. Birkholz c, S. Koke c, G. Steinmeyer c a Johannes Kepler University, Linz, Austria b Chemnitz University of Technology, Germany c Max Born Institute, Berlin, Germany Doctoral Program Computational Mathematics Numerical Analysis and Symbolic Computation Shanghai, 212-1-19 Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 1 / 37
Introduction SD-SPIDER method Mathematical Analysis Discretization Regularization Numerical results Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 2 / 37
Introduction Overview Introduction SD-SPIDER method Mathematical Analysis Discretization Regularization Numerical results Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 2 / 37
Introduction Motivation Why study ultra-short laser pulses? to create shorter, stronger pulses; to enhance optical systems; medicine, material processing, etc. Problem: measurements limited by electronics (order 1 12 s) Development of pulse durations: Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 3 / 37
Introduction Motivation Why study ultra-short laser pulses? to create shorter, stronger pulses; to enhance optical systems; medicine, material processing, etc. Problem: measurements limited by electronics (order 1 12 s) Development of pulse durations: Solution: sample pulse by itself Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 3 / 37
Introduction Laser pulse representation Time domain: electric field E(t), envelope A(t), intensity I(t) = A(t) 2 a) A(t) b) FT A(ω) φ(ω) E(t) Fourier domain: amplitude A(ω), phase ϕ(ω), spectrum I(ω) = A(ω) 2 Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 4 / 37
SD-SPIDER method Overview Introduction SD-SPIDER method Mathematical Analysis Discretization Regularization Numerical results Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 4 / 37
SD-SPIDER method SD-SPIDER= Self-Defraction Spectral Phase Interferometry for Direct Electric-field Reconstruction Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 5 / 37
SD-SPIDER method SD-SPIDER= Self-Defraction Spectral Phase Interferometry for Direct Electric-field Reconstruction introduced by the research group Solid State Light Sources led by Dr. Günter Steinmeyer as subdivision of division C Nonlinear Processes in Condensed Matter at Max-Born-Institute for Nonlinear Optics and Short Pulse Spectroscopy, Berlin, Germany Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 5 / 37
SD-SPIDER method SD-SPIDER= Self-Defraction Spectral Phase Interferometry for Direct Electric-field Reconstruction introduced by the research group Solid State Light Sources led by Dr. Günter Steinmeyer as subdivision of division C Nonlinear Processes in Condensed Matter at Max-Born-Institute for Nonlinear Optics and Short Pulse Spectroscopy, Berlin, Germany theory presented at Conference on Lasers and Electro-Optics, 21 Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 5 / 37
SD-SPIDER method SD-SPIDER= Self-Defraction Spectral Phase Interferometry for Direct Electric-field Reconstruction introduced by the research group Solid State Light Sources led by Dr. Günter Steinmeyer as subdivision of division C Nonlinear Processes in Condensed Matter at Max-Born-Institute for Nonlinear Optics and Short Pulse Spectroscopy, Berlin, Germany theory presented at Conference on Lasers and Electro-Optics, 21 reasons for introduction: applicable for ultraviolet radiation, good signal strength because it uses third-order optical effects Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 5 / 37
SD-SPIDER method basics of nonlinear optics Polarization P caused by an electric field Ẽ, P (t) = ɛ [χ (1) Ẽ(t) + χ (2) Ẽ 2 (t) + χ (3) Ẽ 3 (t) +... ] may act as source of electromagnetic radiation: ( E) + n2 c 2 2 t E = µ 2 t P NL (E) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 6 / 37
SD-SPIDER method basics of nonlinear optics Polarization P caused by an electric field Ẽ, P (t) = ɛ [χ (1) Ẽ(t) + χ (2) Ẽ 2 (t) + χ (3) Ẽ 3 (t) +... ] may act as source of electromagnetic radiation: ( E) + n2 c 2 2 t E = µ 2 t P NL (E) third-order term dominant: χ (3) -medium Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 6 / 37
SD-SPIDER method basics of nonlinear optics Polarization P caused by an electric field Ẽ, P (t) = ɛ [χ (1) Ẽ(t) + χ (2) Ẽ 2 (t) + χ (3) Ẽ 3 (t) +... ] may act as source of electromagnetic radiation: ( E) + n2 c 2 2 t E = µ 2 t P NL (E) third-order term dominant: χ (3) -medium Refraction index n and Kerr-effect: n(ω) = n + n 2 E(ω) 2, (each frequency is refracted slightly differently) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 6 / 37
SD-SPIDER method χ (3) -media allow a four-wave mixing process Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 7 / 37
SD-SPIDER method Principle Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 8 / 37
SD-SPIDER method k-vector-diagram: k - k cw k cw X (3) k (-) SD k p k p k p (+) k SD k cw kcw k -k p k(ω SD, ω p, ω cw ) = k cw (ω cw ) + k p (ω p ) + k p (ω SD + ω cw ω p ) k SD (ω SD, ω cw, ω p ). Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 9 / 37
SD-SPIDER method k-vector-diagram: k - k cw k cw X (3) k (-) SD k p k p k p (+) k SD k cw kcw k -k p k(ω SD, ω p, ω cw ) = k cw (ω cw ) + k p (ω p ) + k p (ω SD + ω cw ω p ) k SD (ω SD, ω cw, ω p ). energy conservation ω p + ω p = ω SD + ω cw still holds Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 9 / 37
SD-SPIDER method The autoconvolution effect pulses considered as plane waves: cw p 2 p 1 Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 1 / 37
SD-SPIDER method The autoconvolution effect pulses considered as plane waves: cw p 2 p 1 interference pattern creates refractive index grating (Kerr-effect) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 1 / 37
SD-SPIDER method The autoconvolution effect pulses considered as plane waves: cw p 2 p 1 interference pattern creates refractive index grating (Kerr-effect) a wave p 1 of each frequency creates an interference pattern with cw-wave Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 1 / 37
SD-SPIDER method The autoconvolution effect pulses considered as plane waves: cw p 2 p 1 interference pattern creates refractive index grating (Kerr-effect) a wave p 1 of each frequency creates an interference pattern with cw-wave at each pattern, photons p 2 of each frequency are refracted Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 1 / 37
SD-SPIDER method The autoconvolution effect pulses considered as plane waves: cw p 2 p 1 interference pattern creates refractive index grating (Kerr-effect) a wave p 1 of each frequency creates an interference pattern with cw-wave at each pattern, photons p 2 of each frequency are refracted SD-signal is sum of all combinations E p (ω p )E p (ω SD + ω cw ω p )E cw Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 1 / 37
SD-SPIDER method Equation in physical formulation E SD (ω SD ) = ω SD +ω cw K(ω SD, ω p )E p (ω p )E p (ω SD + ω cw ω p )dω p supp E p = [ω l p, ω u p ], supp E SD = [2ω l p ω cw, 2ω u p ω cw ], Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 11 / 37
SD-SPIDER method Equation in physical formulation E SD (ω SD ) = ω SD +ω cw K(ω SD, ω p )E p (ω p )E p (ω SD + ω cw ω p )dω p supp E p = [ω l p, ω u p ], supp E SD = [2ω l p ω cw, 2ω u p ω cw ], with kernel K(ω SD, ω p ) = µ cl 2 ω SD K continuous, complex valued n(ω SD ) χ(3) (ω SD, ω cw, ω p, ω SD + ω cw ω p ) E cw e i( k ξ ξ+ k ηη+ k ζ L 2 ) sinc( k ζ L 2 ) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 11 / 37
SD-SPIDER method Equation in physical formulation E SD (ω SD ) = ω SD +ω cw K(ω SD, ω p )E p (ω p )E p (ω SD + ω cw ω p )dω p supp E p = [ω l p, ω u p ], supp E SD = [2ω l p ω cw, 2ω u p ω cw ], with kernel K(ω SD, ω p ) = µ cl 2 ω SD K continuous, complex valued unknown, so far neglected n(ω SD ) χ(3) (ω SD, ω cw, ω p, ω SD + ω cw ω p ) E cw e i( k ξ ξ+ k ηη+ k ζ L 2 ) sinc( k ζ L 2 ) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 11 / 37
SD-SPIDER method mathematical formulation after transformation and renaming: s y(s) = F [x](s) = k(s, t)x(t)x(s t)dt y = F (x) t 1, s 2 Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 12 / 37
SD-SPIDER method mathematical formulation after transformation and renaming: y(s) = F [x](s) = s k(s, t)x(t)x(s t)dt y = F (x) t 1, s 2 x L 2 C [, 1], y L2 C [, 2], k L2 C ([, 2] [, 1]) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 12 / 37
SD-SPIDER method mathematical formulation after transformation and renaming: y(s) = F [x](s) = s k(s, t)x(t)x(s t)dt y = F (x) t 1, s 2 x L 2 C [, 1], y L2 C [, 2], k L2 C ([, 2] [, 1]) fundamental pulse: x(t) = A(t)e iϕ(t) measured SD-pulse: y(s) = B(s)e iψ(s) available Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 12 / 37
SD-SPIDER method mathematical formulation after transformation and renaming: y(s) = F [x](s) = s k(s, t)x(t)x(s t)dt y = F (x) t 1, s 2 x L 2 C [, 1], y L2 C [, 2], k L2 C ([, 2] [, 1]) fundamental pulse: x(t) = A(t)e iϕ(t) measured SD-pulse: y(s) = B(s)e iψ(s) available, possibly available Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 12 / 37
SD-SPIDER method mathematical formulation after transformation and renaming: y(s) = F [x](s) = s k(s, t)x(t)x(s t)dt y = F (x) t 1, s 2 x L 2 C [, 1], y L2 C [, 2], k L2 C ([, 2] [, 1]) fundamental pulse: x(t) = A(t)e iϕ(t) measured SD-pulse: y(s) = B(s)e iψ(s) available, possibly available, unknown ϕ(t) = ϕ + t GD(τ)dτ Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 12 / 37
SD-SPIDER method Does B(s) provide important information? 4 Phase I 4 Phase II 2 2 2 2 4 4 6 6 8 25 3 35 4 45 5 55 6 frequency (THz) 8 25 3 35 4 45 5 55 6 frequency (THz) 5 convolved phase I 5 convolved phase II 5 5 1 15 2 1 15 2 25 25 3 3 1 2 3 4 5 6 7 8 frequency (THz) 35 1 2 3 4 5 6 7 8 frequency (THz) 1 convolved absolute values I 1 convolved absolute values II.8.8.6.6.4.4.2.2 1 2 3 4 5 6 7 8 frequency (THz) 1 2 3 4 5 6 7 8 frequency (THz) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 13 / 37
SD-SPIDER method Does B(s) provide important information? 4 Phase I 4 Phase II 2 2 2 2 4 4 6 6 8 25 3 35 4 45 5 55 6 frequency (THz) 8 25 3 35 4 45 5 55 6 frequency (THz) 5 convolved phase I 5 convolved phase II 5 5 1 15 2 1 15 2 25 25 3 3 1 2 3 4 5 6 7 8 frequency (THz) 35 1 2 3 4 5 6 7 8 frequency (THz) 1 convolved absolute values I 1 convolved absolute values II.8.8.6.6.4.4.2.2 1 2 3 4 5 6 7 8 frequency (THz) 1 2 3 4 5 6 7 8 frequency (THz) Yes, it does! Thus also B(s) available as measurement. Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 13 / 37
SD-SPIDER method measurements (indicated by δ) close to correct data, but not exact A δ A, B δ B, ψ δ ψ as δ Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 14 / 37
SD-SPIDER method measurements (indicated by δ) close to correct data, but not exact A δ A, B δ B, ψ δ ψ as δ no information about size of error δ available Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 14 / 37
SD-SPIDER method measurements (indicated by δ) close to correct data, but not exact A δ A, B δ B, ψ δ ψ as δ no information about size of error δ available Statement of the problem: given A δ, B δ, ψ δ and k(s, t), find ϕ such that B δ (s)e iψδ (s) = s k(s, t)a δ (t)e iϕ(t) A δ (s t)e iϕ(s t) dt Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 14 / 37
Mathematical Analysis Overview Introduction SD-SPIDER method Mathematical Analysis Discretization Regularization Numerical results Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 14 / 37
Mathematical Analysis Ill-posedness F x = y, F : L 2 [, 1] L 2 [, 2] An operator F is called ill-posed, if it violates at least one of Hadamard s conditions: (a) for each given data y there exists a solution x (b) this solution is unique (c) the solution depends continuously on the data Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 15 / 37
Mathematical Analysis Ill-posedness F x = y, F : L 2 [, 1] L 2 [, 2] An operator F is called ill-posed, if it violates at least one of Hadamard s conditions: (a) for each given data y there exists a solution x (b) this solution is unique (c) the solution depends continuously on the data (a) violated because F (x) C C [, 2] x L 2 C [, 1] Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 15 / 37
Mathematical Analysis Injectivity for k(s, t) 1 and k(s, t) = k(s): F (x 1 ) = F (x 2 ) has two solutions x 1 = x 2 and x 1 = x 2 by Titchmarsh s theorem Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 16 / 37
Mathematical Analysis Injectivity for k(s, t) 1 and k(s, t) = k(s): F (x 1 ) = F (x 2 ) has two solutions x 1 = x 2 and x 1 = x 2 by Titchmarsh s theorem for k(s, t) again x 1 = x 2 or x 1 = x 2, additional solutions are an open problem. Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 16 / 37
Mathematical Analysis Injectivity for k(s, t) 1 and k(s, t) = k(s): F (x 1 ) = F (x 2 ) has two solutions x 1 = x 2 and x 1 = x 2 by Titchmarsh s theorem for k(s, t) again x 1 = x 2 or x 1 = x 2, additional solutions are an open problem. (b) is violated too! Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 16 / 37
Mathematical Analysis Injectivity for k(s, t) 1 and k(s, t) = k(s): F (x 1 ) = F (x 2 ) has two solutions x 1 = x 2 and x 1 = x 2 by Titchmarsh s theorem for k(s, t) again x 1 = x 2 or x 1 = x 2, additional solutions are an open problem. (b) is violated too! but since x 1 = Ae iϕ, x 1 = x 2 means x 2 = Ae i(ϕ π) and both solutions are equivalent for our problem. Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 16 / 37
Mathematical Analysis Injectivity for k(s, t) 1 and k(s, t) = k(s): F (x 1 ) = F (x 2 ) has two solutions x 1 = x 2 and x 1 = x 2 by Titchmarsh s theorem for k(s, t) again x 1 = x 2 or x 1 = x 2, additional solutions are an open problem. (b) is violated too! but since x 1 = Ae iϕ, x 1 = x 2 means x 2 = Ae i(ϕ π) and both solutions are equivalent for our problem. because of periodicity, ϕ ϕ + 2π Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 16 / 37
Mathematical Analysis (local) ill-posedness for the autoconvolution operator, compactness can not be proven in general nonlinear operator requires local analysis Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 17 / 37
Mathematical Analysis (local) ill-posedness for the autoconvolution operator, compactness can not be proven in general nonlinear operator requires local analysis Definition We define an operator F, F : X Y to be locally ill-posed in x X if, for arbitrarily small ρ > there exists a sequence {x n } B ρ (x ) X satisfying the condition F(x n ) F(x ) in Y as n, but x n x in X. Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 17 / 37
Mathematical Analysis (local) ill-posedness for the autoconvolution operator, compactness can not be proven in general nonlinear operator requires local analysis Definition We define an operator F, F : X Y to be locally ill-posed in x X if, for arbitrarily small ρ > there exists a sequence {x n } B ρ (x ) X satisfying the condition F(x n ) F(x ) in Y as n, but x n x in X. Theorem (Gorenflo & Hofmann 94, adapted in Gerth 11) The autoconvolution operator F is everywhere locally ill-posed. (c) is violated too! Regularization is necessary. Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 17 / 37
Mathematical Analysis Fréchet-derivative The Fréchet-derivative of F in a point x is given by s [F (x )h](s) = (k(s, t) + k(s, s t))x (s t)h(t)dt Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 18 / 37
Mathematical Analysis Fréchet-derivative The Fréchet-derivative of F in a point x is given by s [F (x )h](s) = (k(s, t) + k(s, s t))x (s t)h(t)dt although F is in general non-compact, F is always compact! Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 18 / 37
Discretization Overview Introduction SD-SPIDER method Mathematical Analysis Discretization Regularization Numerical results Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 18 / 37
Discretization equation: y(s) = s k(s, t)x(s t)x(t)dt supp x = [t l, t u ], supp y = [2t l t cw, 2t u t cw ] Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 19 / 37
Discretization equation: y(s) = s k(s, t)x(s t)x(t)dt supp x = [t l, t u ], supp y = [2t l t cw, 2t u t cw ] discretization using rectangular rule y(s m ) = N k(s m, t j )x(s m + t cw t j )x(t j ) t j=1 with t = tu t l N 1, t j = t l + (j 1) t, s m = 2t j + (m 1) t y m := y(s m ), x n := x(t n ), k m,n := k(s m, t n ) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 19 / 37
Discretization in matrix-form y = F (x)x, with Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 2 / 37
Discretization in matrix-form y = F (x)x, with y/ t = F x/ t = k 1,1 x 1... k 2,1 x 2 k 2,2 x 1.......... k N 1,1 x N 1 k N 1,2 x N 2... k N 1,N 1 x 1 k N,1 x N k N,2 x N 1... k N,N 1 x 2 k N,N x 1 k N+1,1 x N... k N+1,N 1 x 3 k N+1,N 1 x 2.......... k 2N 2,N 1 x N k 2N 2,N x N 1... k 2N 1,N x N x 1 x 2. x N 1 x N Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 2 / 37
Discretization in matrix-form y = F (x)x, with y/ t = F x/ t = k 1,1 x 1... k 2,1 x 2 k 2,2 x 1.......... k N 1,1 x N 1 k N 1,2 x N 2... k N 1,N 1 x 1 k N,1 x N k N,2 x N 1... k N,N 1 x 2 k N,N x 1 k N+1,1 x N... k N+1,N 1 x 3 k N+1,N 1 x 2.......... k 2N 2,N 1 x N k 2N 2,N x N 1... k 2N 1,N x N Decomposition, with as element-by-element multiplication: F = K X x 1 x 2. x N 1 x N Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 2 / 37
Discretization analogously: Fréchet-derivative N [F (x )h] m = (k(s m, t j )+k(s m, s m +t cw t j ))x (s m +t cw t j )h(t j ) t j=1 Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 21 / 37
Discretization analogously: Fréchet-derivative N [F (x )h] m = (k(s m, t j )+k(s m, s m +t cw t j ))x (s m +t cw t j )h(t j ) t j=1 resulting matrix F (x ) = (K + K ) X Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 21 / 37
Discretization analogously: Fréchet-derivative [F (x )h] m = N (k(s m, t j )+k(s m, s m +t cw t j ))x (s m +t cw t j )h(t j ) t j=1 resulting matrix F (x ) = (K + K ) X advantage: time-consuming calculation of the matrices K and K has to be performed only once for each measurement setup Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 21 / 37
Regularization Overview Introduction SD-SPIDER method Mathematical Analysis Discretization Regularization Numerical results Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 21 / 37
Regularization A Levenberg-Marquardt-Type approach we let the complete pulse x be unknown, whereas y is given Iteration rule: 1 x δ (l+1) := xδ (l) (F +γ (x δ (l) ) F (x δ (l) L) ) + αl F (x δ (l) ) (y δ F (x δ (l) ) for l =,..., l, aimed at minimizing y δ F (x (l) ) F (x (l) )(x x (l) ) 2 + α L(x x (l) ) 2, L(x) approximating the second derivative of x Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 22 / 37
Regularization A Levenberg-Marquardt-Type approach we let the complete pulse x be unknown, whereas y is given Iteration rule: x δ (l+1) := xδ (l) +γ (F (x δ (l) ) F (x δ (l) ) + αl L) 1 F (x δ (l) ) (y δ F (x δ (l) ) for l =,..., l, aimed at minimizing y δ F (x (l) ) F (x (l) )(x x (l) ) 2 + α L(x x (l) ) 2, L(x) approximating the second derivative of x Questions: how to choose x? how to choose l? how to choose α? Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 22 / 37
Regularization Choice of x = A e iϕ obviously, A := A δ first idea for phase: ϕ (t) x 1 28 Spectr. Pow. Dens. ( x(t) ) 1.5 1.5 original pulse starting phase reconstructed pulse 25 3 35 4 45 5 55 6 3 phase (arg(x(t))) 2 1 1 25 3 35 4 45 5 55 6 Frequency (THz) (δ =, α = ) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 23 / 37
Regularization idea: calculate good guess. Observe B δ (s)e iψδ (s) = s k(s, t) A δ (t)a δ (s t)e i(ϕ(t)+ϕ(s t)+φ kernel) dt set ϕ (t) = 1 2 (P s t(ψ(s))) φ kernel (s, t) for s fixed Spectr. Pow. Dens. ( x(t) ) 1.2 x 1 28 1.8.6.4.2 original pulse starting phase reconstructed pulse 25 3 35 4 45 5 55 6 1 phase (arg(x(t))) 5 5 1 15 25 3 35 4 45 5 55 6 Frequency (THz) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 24 / 37
Regularization problem for slightly changed fundamental phase 8 6 4 phase (arg(x(t))) 2 2 4 6 25 3 35 4 45 5 55 6 Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 25 / 37
Regularization best result with kernel correction Spectr. Pow. Dens. ( x(t) ) 2.5 x 1 28 2 1.5 1.5 original pulse starting phase reconstructed pulse 25 3 35 4 45 5 55 6 3 phase (arg(x(t))) 2 1 1 25 3 35 4 45 5 55 6 Frequency (THz) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 26 / 37
Regularization best result with kernel correction Spectr. Pow. Dens. ( x(t) ) 2.5 x 1 28 2 1.5 1.5 original pulse starting phase reconstructed pulse 25 3 35 4 45 5 55 6 3 phase (arg(x(t))) 2 1 1 25 3 35 4 45 5 55 6 Frequency (THz) set starting phase to constant zero Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 26 / 37
Regularization When to stop the iteration? An example iteration: (l) F (x δ (l) ) yδ x δ (l) Aδ 1 9.5819e-1.5252 2 2.4115e-2.7916 4 2.682e-2.7937 6 1.5369e-2.677 1 1.3792e-3.1964 12 1.122e-3.171 14 9.4595e-4.1623 143 9.234e-4.1622 144 9.166e-4.1623 15 8.748e-4.1632 25 3.1613e-4.22 Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 27 / 37
Regularization When to stop the iteration? An example iteration: (l) F (x δ (l) ) yδ x δ (l) Aδ 1 9.5819e-1.5252 2 2.4115e-2.7916 4 2.682e-2.7937 6 1.5369e-2.677 1 1.3792e-3.1964 12 1.122e-3.171 14 9.4595e-4.1623 143 9.234e-4.1622 144 9.166e-4.1623 15 8.748e-4.1632 25 3.1613e-4.22 choose l such that x δ (l) Aδ is minimal Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 27 / 37
Regularization Choice of α no a-priori information y y δ < δ available, thus a-posteriori methods necessary Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 28 / 37
Regularization Choice of α no a-priori information y y δ < δ available, thus a-posteriori methods necessary calculate solutions for various α, e.g. α n = α q n, < q < 1, n =,..., n max and take best solution Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 28 / 37
Regularization Choice of α no a-priori information y y δ < δ available, thus a-posteriori methods necessary calculate solutions for various α, e.g. α n = α q n, < q < 1, n =,..., n max and take best solution L-curve not applicable, quasioptimality ( x αi+1 x αi min) failed Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 28 / 37
Regularization Choice of α no a-priori information y y δ < δ available, thus a-posteriori methods necessary calculate solutions for various α, e.g. α n = α q n, < q < 1, n =,..., n max and take best solution L-curve not applicable, quasioptimality ( x αi+1 x αi min) failed instead, make use of A δ again: choose α such that x δ α Aδ = min n x δ α n A δ Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 28 / 37
Numerical results Overview Introduction SD-SPIDER method Mathematical Analysis Discretization Regularization Numerical results Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 28 / 37
Numerical results A very smooth fundamental pulse 1.4 x 1 7 absolute values 6 phase 1.2 1 5 4 3.8.6 2 1.4.2 1 2 3 25 3 35 4 45 5 55 6 frequency (THz) 4 25 3 35 4 45 5 55 6 frequency (THz) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 29 / 37
Numerical results SD-pulse, 5% relative noise added 2.5 x 114 absolute values 12 phase 2 1 1.5 8 6 1 4.5 2 1 2 3 4 5 6 7 8 frequency (THz) 1 2 3 4 5 6 7 8 frequency (THz) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 3 / 37
Numerical results reconstruction, α = 5.86 1 6 1.4 x absolute values 1 7 1.2 original reconstructed 6 5 phase original reconstructed 4 1 3.8 2 1.6.4 1 2.2 3 25 3 35 4 45 5 55 6 frequency (THz) 4 25 3 35 4 45 5 55 6 frequency (THz) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 31 / 37
Numerical results A more oscillating pulse 1.4 x 1 7 absolute values 8 phase 1.2 6 1 4.8 2.6.4 2.2 4 25 3 35 4 45 5 55 6 frequency (THz) 6 25 3 35 4 45 5 55 6 frequency (THz) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 32 / 37
Numerical results noise-free SD-pulse 12 x 113 absolute values 1 phase 1 8 5 5 6 1 4 2 15 2 25 1 2 3 4 5 6 7 8 frequency (THz) 3 1 2 3 4 5 6 7 8 frequency (THz) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 33 / 37
Numerical results reconstruction, α = 2.17 1.4 x 1 7 absolute values 1.2 original reconstructed 8 6 phase original reconstructed 1 4.8 2.6.4 2.2 4 25 3 35 4 45 5 55 6 frequency (THz) 6 25 3 35 4 45 5 55 6 frequency (THz) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 34 / 37
Numerical results reconstruction, 1% relative noise in data 1.4 x 1 7 absolute values 1.2 original reconstructed 8 6 phase original reconstructed 1 4.8 2.6.4 2.2 4 25 3 35 4 45 5 55 6 frequency (THz) 6 25 3 35 4 45 5 55 6 frequency (THz) Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 35 / 37
Numerical results Real data situation unfortunately, no results available. Main reasons: measurements without magnitudes unknown factor in model error in the model frequency domains of x and y do not match Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 36 / 37
Numerical results D. Gerth, B. Hofmann, S. Birkholz, S. Koke, and G. Steinmeyer Regularization of an autoconvolution problem in ultrashort laser pulse characterization, submitted D. Gerth, Regularization of an autoconvolution problem occurring in measurements of ultra-short laser pulses, Diploma thesis, Chemnitz University of Technology, Chemnitz, 211, http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-85485. R. Gorenflo, B. Hofmann, On autoconvolution and regularization, Inverse Problems 1 (1994), pp. 353 373. S. Koke, S. Birkholz, J. Bethge, C. Grebing, G. Steinmeyer, Self-diffraction SPIDER, Conference on Laser and Electro Optics (CLEO), San Jose, CA, 28. Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 37 / 37
Numerical results D. Gerth, B. Hofmann, S. Birkholz, S. Koke, and G. Steinmeyer Regularization of an autoconvolution problem in ultrashort laser pulse characterization, submitted D. Gerth, Regularization of an autoconvolution problem occurring in measurements of ultra-short laser pulses, Diploma thesis, Chemnitz University of Technology, Chemnitz, 211, http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-85485. R. Gorenflo, B. Hofmann, On autoconvolution and regularization, Inverse Problems 1 (1994), pp. 353 373. S. Koke, S. Birkholz, J. Bethge, C. Grebing, G. Steinmeyer, Self-diffraction SPIDER, Conference on Laser and Electro Optics (CLEO), San Jose, CA, 28. Thank you for your attention! Are there any questions? Gerth, Hofmann, Birkholz, Koke, Steinmeyer JKU/TUC/MBI 37 / 37