Extending time-domain ptychography to generalized phase-only transfer functions

Extending time-domain pt yc hograph y to generalized phase-only transfer functions Dirk-Math ys Spangen b erg, 1 , ∗ Eric h Roh wer, 1 Mic hael Br ¨ ugmann, 2 and Thomas F eurer 2 1 L aser R ese ar ch Institute, Stel lenb osch University, Private Bag X1, 7602 Matieland, South Afric a 2 Institute of Applie d Physics, University of Bern, Sid lerstr asse 5, 3012 Bern, Switzerland (Dated: April 19, 2019) W e extend the time-domain ptyc hographic iterativ e engine to generalized spectral phase-only transfer functions. The mo dified algorithm, i 2 PIE, is describ ed and its robustness is demonstrated b y differen t n umeric simulations. The concept is experimentally verified by reconstruction of a complex supercontin uum pulse from an all normal dispersion fib er. P A CS n um b ers: 42.30.-d, 42.30.Rx, 42.65.Re Reco very of coherent broadband signals, esp ecially their phase, is a con tin uously dev eloping field as detec- tors are often not fast enough to make direct measure- men ts. Recently ptyc hography , a robust lens-less imag- ing tec hnique dev elop ed to solv e the phase problem in crystallograph y by Hopp e [1], has b een migrated to the time domain [2] b y application of the ptyc hographic itera- tiv e engine (PIE) [3] to time domain equiv alent problems. Time-domain ptyc hograph y requires the measurement of a sp ectrum resulting from the product of t wo coheren t signals, the unkno wn ob ject and a time delay ed probe signal. This is done for a num b er of time dela ys resulting in a sequence of sp ectra whic h is referred to as a sp ectro- gram. In the most fundamental version the spectrogram is fed to the ptyc hographic iterative engine which recon- structs the unknown ob ject signal given that the prob e signal is kno wn. Refined co des, e.g., the extended pt ycho- graphic iterativ e engine (ePIE), make use of redundancy in the sp ectrogram to also reconstruct the probe signal [4]. More recently , a new modality , i.e., the implicit pt y- c hographic iterative engine (iPIE), was introduced [5, 6]. In iPIE the sp ectrogram is generated from the pro duct of an unkno wn ob ject with a prob e signal whic h is derived from the ob ject by application of a linear sp ectral transfer function. Even though time-domain ptyc hograph y can b e applied to all coherent broadband signals irresp ective of carrier frequency , exp erimen ts published up to now fo- cused on ultrafast broadband laser pulses. F or example, PIE has b een shown to reliably reconstruct unknown ul- trafast pulses from corresp onding sp ectrograms or cross- correlation frequency resolv ed optical gating (XFR OG) traces [2, 7]. In this work, we extend pt ychograph y to reconstruct unkno wn ob ject signals entirely without a prob e signal, but by application of different families of sp ectral phase- only transfer functions. W e call the scheme i 2 PIE since w e measure the squar e of a signal whic h is the result of applying families of kno wn transfer functions, i.e., in- trinsic know le dge , to the ob ject signal. The new i 2 PIE sc heme has the p oten tial to simplify ultrafast pulse re- construction as no prob e pulse is required. Instead, it analyzes sp ectra whic h come from collinear second harmonic generation of phase modulated ob ject pulses. Th us, p ossible exp erimen tal arrangemen ts are similar to those used in m ultiphoton intrapulse interference phase scan (MI IPS) [8], interferometric frequency resolved op- tical gating (iFROG) [9, 10], shaper assisted collinear sp ectral phase in terferometry for direct electric field re- construction (SPIDER) [11] or the disp ersion scan (D- Scan) method [12]. Exp erimentally , w e implement the metho d using a 4f-shap er with a n SLM to apply selected sp ectral transfer functions and reconstruct a complex su- p ercon tin uum pulse from an all normal disp ersion fib er. The i 2 PIE algorithm takes a sp ectrogram as input. The sp ectrogram S (Ω , n ), consisting of n measured sp ec- tra S n (Ω), is recorded by applying each transfer function H n (Ω) from a set of known spectral phase-only trans- fer functions H (Ω , n ) sequen tially to the unknown ob ject signal and recording the resultant second harmonic sp ec- trum. Here Ω = ω − ω 0 is defined relative to the carrier frequency ω 0 . More formally , for each transfer function in a set, the pro duct of the transfer function H n (Ω) with the ob ject pulse E in (Ω), o n (Ω) = E in (Ω) H n (Ω) , (1) is sent into a nonlinear mixer and the resultan t sp ec- trally resolved second harmonic intensit y is recorded, S n (Ω) =   F  o 2 n ( t )    2 , (2) where F denotes the F ourier transformation. F rom suc h a sp ectrogram the unknown ob ject signal E in can be reconstructed using the i 2 PIE algorithm as follows. An initial guess is made for the ob ject signal E 0 in (Ω) which defines the mo dulated signal o n (Ω) based on the corre- sp onding transfer function, o n (Ω) = E 0 in (Ω) H n (Ω) . (3) Assuming p erfect phase matching o v er the entire spec- tral bandwidth, the second harmonic signal is g n ( t ) = o 2 n ( t ) . (4) 2 This second harmonic signal is used to calculate an up- dated field g 0 n b y replacing the current estimated ampli- tude with the measured amplitude from the corresp ond- ing sp ectrum S n (Ω), i.e., g 0 n (Ω) = p S n (Ω) exp[i arg( g n (Ω))] . (5) No w the mo dulated signal is up dated follo wing stan- dard ptyc hographic recip e o 0 n ( t ) = o n ( t ) + β U n ( t ) [ g 0 n ( t ) − g n ( t )] (6) where U ( t ) = | o n ( t ) | max ( | o n ( t ) | ) o ∗ n ( t ) | o n ( t ) 2 | + α . (7) W e use a constant weigh t β ∈ [0 . . . 1] and α < 1. The last step, unique to the i 2 PIE algorithm, is to up date the curren t estimate of the ob ject signal E 0 in , E 0 in (Ω) = o 0 n (Ω) H ∗ n (Ω) , (8) using the intrinsic kno wledge of the transfer function used and the current up dated second harmonic signal o 0 n . The pro cedure is rep eated for all recorded spectra m ultiple times until the ob ject signal E in is sufficien tly reconstructed. As with other v ariants of PIE there is a redundancy of information requirement. This is ac hieved by ha v- ing a sufficient num b er of transfer functions and sensible c hoices for the sp ecific type of transfer functions. Here, w e organize transfer functions in to families with the same basis function and we sho w ho w one can calculate sensi- ble boundary v alues for the free parameters of a trans- fer function family . More formally , to reconstruct the slo wly v arying en velope of an unkno wn ob ject signal, us- ing i 2 PIE, E in (Ω) = A (Ω) e i ψ (Ω) (9) a family of phase-only transfer functions [ ψ n (Ω)] with n ∈ [1 . . . N ] is chosen. W e restrict ourselves to families that can b e characterized by only a few parameters and w e will discuss t wo examples, i.e. families of p olynomial and sinusoidal phases. The parameter b oundaries of the chosen transfer func- tion are either given b y the experimental setup, or fun- damen tally , by discrete sampling theory . In the latter case they are just as easily obtainable for a measure- men t as they are for the sim ulated signals. Besides the transfer functions, we assume to know the sp ectral reso- lution ∆Ω of the sp ectrometer and the sp ectrum of the unkno wn ob ject signal I (Ω) = | A (Ω) | 2 . F rom the sp ec- trometer res olution we calculate the total time window, i.e. T = 2 π / ∆Ω. F urther, w e assume that the maxi- m um applied transfer function ψ N (Ω) dominates the to- tal phase, i.e. ψ tot (Ω) = ψ (Ω) + ψ N (Ω) ≈ ψ N (Ω). With this we can estimate the ob ject signal duration after ap- plying the maximum transfer function ψ N (Ω) to σ 2 t = 1 2 π ∞ Z −∞ dΩ (  ∂ A (Ω) ∂ Ω  2 +  ∂ ψ N (Ω) ∂ Ω + t  A (Ω)  2 ) (10) with t , the first moment of the temp oral intensit y . While the first term on the righ t hand side represents the bandwidth limited duration σ 2 0 . = 1 2 π Z dΩ  ∂ A (Ω) ∂ Ω  2 , (11) the second term describ es signal broadening due to the applied phase mo dulation. Hereafter, we assume t = 0 which is approximately true for most cases dis- cussed here. W e determine the parameter b oundaries of ψ N (Ω) by restricting the duration of the mo dulated ob- ject signal to a fraction γ of the total time window, i.e. γ T . Therefore, broadening due to ψ N (Ω) should at most b e equal to σ ψ = q γ 2 T 2 − σ 2 0 (12) Consider the tw o families of transfer functions dis- cussed hereafter. First, the family of p olynomial phases ψ (Ω) = ± q Ω k (13) with parameter q and constant order k ≥ 2. With eq. (10) we find for the maximum allo wed k -th order phase q max = ± s σ 2 ψ k 2 2 π R dΩ Ω 2( k − 1) I (Ω) (14) whic h can b e easily calculated knowing γ T and the ob ject sp ectrum I (Ω). Second, we consider the family of sin usoidal phase functions ψ (Ω) = a cos(Ω τ + φ ) (15) The mem b ers of this family are parameterized through amplitude a , frequency τ and phase φ . With eq. (10) we find 3 a max τ max = s 2 σ 2 ψ G (0) − <{ G (2 τ max )e 2i φ } ≈ s 2 σ 2 ψ G (0) (16) with the F ourier transform of the sp ectral intensit y G ( t ) . = 1 2 π Z dΩ I (Ω) e iΩ t . (17) T ypically , the av erage sp ectral intensit y G (0) is larger than <{ G (2 τ max )e 2i φ } and the approximate expres- sion (16) can b e readily used. Also note that the approx- imate expression is indep enden t of the phase φ . That is, w e fix either a max or τ max and use eq (16) to calculate the other. W e ev aluate the i 2 PIE algorithm by reconstruction of a set of random laser pulses. The pulses are used to n u- merically calculate input sp ectrograms based on a trans- fer function family , which are fed to i 2 PIE. In each case w e analyze how well the c hosen transfer function fam- ily performs b y testing its p erformance against the set of random ob ject pulses. F or eac h ob ject pulse we cal- culate the ro ot mean square (rms) error b et ween input and reconstructed sp ectrogram. Reconstructions where log 10 (rms) < − 3 . 5 are considered successful and where log 10 (rms) ≥ − 3 . 5 are considered unsuccessful. The random ob ject set consists of 1000 ob ject pulses. Their randomly shap ed sp ectrum is centered around 800 nm with a sp ectral bandwidth b et w een 2 nm and 20 nm. The sp ectral phase is either, in the first case, p olynomial up to fourth order with random coefficients and, in the second case, sin usoidal with random ampli- tude, frequency and phase. Eac h ob ject pulse is gener- ated for a temp oral window of 8 ps, whic h corresponds to a sp ectral resolution of 0.27 nm at 800 nm, and on a grid of 1024 samples. In all reconstructions w e used α = 0 . 0001 and β = 0 . 3 and eac h reconstruction started with an initial guess of a 200 fs F ourier limited Gaus- sian pulse. W e set γ = 1 / 8 and used N transfer func- tions in all families. F or every ob ject pulse we calculate the parameter boundaries of the resp ective transfer func- tions from γ T and the fundamental spectrum according to eq. (14) and (16). W e sequentially applied the i 2 PIE up date for the entire family of transfer functions in a set and rep eat the process 500 times b efore the rms error w as ev aluated. First, w e start with the family of quadratic phase transfer functions ( k = 2). The individual members are c haracterized by q n = ( n − N / 2 − 1) q max . Sho wn in Fig. 1 are histograms of the logarithmic rms error of all reconstructions with N = 6 , 12 , 25 , 50. W e find that with as little as six transfer functions the method suc- cessfully reconstructs 92.7% of all ob jects. The success FIG. 1. Histogram of the logarithmic rms error of reconstruc- tions of the random pulse set for a family of quadratic phase transfer functions with N = 6 (a), N = 12 (b), N = 25 (c) and N = 50 (d) members. Green bars indicate successful and red bars unsuccessful reconstructions. rate increases to close to 100% for N as large as 50 and the mean of the rms error decreases b y several orders of magnitude. W e find similar results for k = 3 , 4. Next we consider the families of sinusoidal phase trans- fer functions. Three families can b e defined based on v arying the parameters a , τ and φ , respectively . First, w e arbitrarily set τ = 300 fs and calculate the corre- sp onding amplitude for every ob ject pulse using equa- tion (16). Then we v ary φ n b et w een 0 and φ N = 2 π in N equidistant steps. Second, we arbitrarily set φ = 0 and τ = 300 fs, and use equation (16) to calculate the maximum amplitude for every ob ject pulse. The individual transfer functions then ha ve amplitudes of a n = ( n − N / 2 − 1) a max . Finally , we arbitrarily set φ = 0 and a = 2 . 7, calculate τ max for every ob ject pulse and v ary the frequency according to τ n = τ max n/ N . Shown in Fig. 2 are histograms of the logarithmic rms errors for 4 FIG. 2. Histogram of the logarithm of the rms error for three families of sinusoidal phase transfer functions. Green bars indicate successful and red bars unsuccessful reconstructions. The blue curv e shows the cumulativ e p ercentage of successful reconstructions. (a) V arying φ from 0 to 2 π for τ = 300 fs and a from equation (16). (b) Fixing φ = 0, τ = 300 fs and v arying a within the limits calculated with equation (16). (c) Fixing φ = 0, a = 2 . 7 and v arying τ within the limits calculated with eq. (16). the three families of sinusoidal phase transfer function. The p ercentage of successful reconstructions is found to b e 95% and higher. In our lab the broadband ob ject pulse to b e char- acterized is generated b y sending a 800 nm seed pulse with 80 fs duration at 80 MHz rep etition rate from a Ti:Sapphire oscillator into an all-normal disp ersion (ANDi) photonic crystal fib er. The fib er output is then compressed by 48 b ounces on a chirped mirror with 160 fs 2 compression p er b ounce. The resulting pulse serv es as the ob ject pulse E in . A 4f-shap er with a Jenop- tic 640d spatial light modulator is used to sequen tially apply all transfer functions H n (Ω) from a sp ecific fam- ily to the ob ject pulse. The output from the 4f-shap er is fo cused onto a 20 µ m thick BBO crystal by a 0.9 NA ob jective after which the frequency doubled light is col- lected and fo cused in to an Av aSp ec-3648 spectrometer with a resolution of 3.92 THz at 400 nm. The set of recorded second harmonic sp ectra is stored in a sp ectro- gram S n (Ω). W e to ok measurements based on families of quadratic and sin usoidal phase transfer functions where w e v aried FIG. 3. Measured sp ectrogram for scanning (a) the quadratic phase b etw een − 2250 fs 2 and 2250 fs 2 , (b) the sinusoidal phase φ b et ween − π and π with a = 15 and τ = 25, (c) the sinusoidal amplitude a b etw een − 30 and 30 with φ = 0 and τ = 25 fs, and (d) the sin usoidal frequency τ b etw een − 100 fs and 100 fs with φ = 0 and a = π . the respective parameters as discussed in the simulation section. In Fig. 3 the measured sp ectrograms are shown for the differen t cases when (a) the quadratic phase is v aried b et ween − 2250 fs 2 and 2250 fs 2 , (b) the sin usoidal phase φ b et w een − π and π with a = 15 and τ = 25, (c) the sin usoidal amplitude a b etw een − 30 and 30 with φ = 0 and τ = 25 fs, and (d) the sinusoidal frequency τ b et w een − 100 fs and 100 fs with φ = 0 and a = π . The sp ectrograms are further used to retrieve amplitude and phase of the ob ject pulse. FIG. 4. The reconstructed sp ectral intensit y is shown in (a) and the phase in (b). Purple: quadratic phase scan; blue: φ scan; red: amplitude a scan; y ellow: frequency τ scan. In Fig. 4(a) w e plot the reconstructed sp ectral intensi- ties for all four families of sp ectral transfer functions on top of each other and in Fig. 4(b) the resp ectiv e recon- structed sp ectral phases. W e find reasonable agreement in the reconstructed sp ectral amplitudes and excellent agreemen t in the retriev ed phases in regions of nonzero amplitude. In summary , we ha ve demonstrated that the i 2 PIE al- 5 gorithm can reconstruct amplitude and phase of an un- kno wn ob ject signal from a measured second harmonic sp ectrogram recorded by applying differen t families of phase-only sp ectral transfer functions with excellent re- sults. In principle, the choice of family is arbitrary and w e deriv e a formalism that allo ws to calculate the scan limits from only the sp ectral resolution of the sp ectrom- eter and the sp ectral intensit y of the ob ject pulse. A CKNOWLEDGMENTS This research w as funded in part through the Swiss National Science F oundation (Grant Number: 200020- 178812/1) as well as the National Researc h F oundation of South Africa (Gran t Num b er: 47793). ∗ Corresp onding author: dspan@sun.ac.za [1] W. Hopp e. Beugung im inhomogenen Prim¨ arstrahlw ellenfeld. I. Prinzip einer Phasenmessung v on Elektronenbeungungsinterferenzen. A cta Crystal lo- gr aphic a Se ction A , 25(4):495–501, Jul 1969. [2] Dirk Spangenberg, Pieter Neethling, Eric h Roh wer, Mic hael H. Br ¨ ugmann, and Thomas F eurer. Time- domain ptyc hograph y . Phys. R ev. A , 91 , 021803, (2015). [3] H. M. L. F aulkner and J. M. Ro denburg. Error tolerance of an iterative phase retriev al algorithm for mo veable il- lumination microscop y . Ultr amicr osc opy , 103 , 153–164 (2005). [4] M. Lucchini, M. Br ¨ ugmann, A. Ludwig, L. Gallmann, U. Keller, and T. F eurer. Ptyc hographic reconstruction of attosecond pulses. Optics Expr ess , 23 , 29502 (2015). [5] D. Spangen b erg, E. Rohw er, M. H. Br ¨ ugmann, and T. F eurer. Ptyc hographic ultrafast pulse reconstruction. Opt. L ett. , 40 , 1002–1005 (2015). [6] D.-M. Spangenberg, M. Br ¨ ugmann, E. Rohw er, and T. F eurer. All-optical implementation of a time-domain pt ychographic pulse reconstruction setup. Appl. Opt. , 55 , 5008–5013 (2016). [7] A. M. Heidt, D.-M. Spangenberg, M. Br ¨ ugmann, E. G. Roh wer, and T. F eurer. Improv ed retriev al of complex sup ercon tin uum pulses from XFROG traces using a pty- c hographic algorithm. Optics L etters , 41 , 4903 (2016). [8] V. V. Lozov oy , I. Pastirk, and M. Dan tus. Multiphoton in trapulse interference iv ultrashort laser pulse spectral phase characterization and compensation. Optics L etters , 29 , 775 (2004). [9] G. Stib enz and G. Steinmey er. Interferometric frequency- resolv ed optical gating. Optics Expr ess , 13 , 2617 (2005). [10] A. Galler and T. F eurer. Pulse shaper assisted short laser pulse characterization. Applie d Physics B , 90 , 427–430 (2008). [11] C. Iaconis and I. A. W almsley . Sp ectral phase in terferom- etry for direct electric-field reconstruction of ultrashort optical pulses. Optics L etters , 23 , 792 (1998). [12] M. Miranda, C. L. Arnold, T. F ordell, F. Silv a, B. Alonso, R. W eigand, A. L’Huillier, and H. Cresp o. Characteriza- tion of broadband few-cycle laser pulses with the d-scan tec hnique. Optics Expr ess , 20 , 18732 (2012).