Measurement of Energy Contrast of Amplified Ultrashort Pulses using Cross Polarized Wave Generation...

Measurement of energy contrast ofamplified ultrashort pulses using

cross-polarized wave generation andspectral interferometry

Marin Iliev,1 Amanda K. Meier,1 Benjamin Galloway,2 Daniel E.Adams,2 Jeff A. Squier,1 and Charles G. Durfee1,∗

1Department of Physics, Colorado School of Mines, Golden, CO 80401, USA2JILA, University of Colorado and National Institute of Standards and Technology, Boulder,

CO 80309, USA∗[email protected]

Abstract: We present a method using spectral interferometry (SI) tocharacterize a pulse in the presence of an incoherent background suchas amplified spontaneous emission (ASE). The output of a regenerativeamplifier is interfered with a copy of the pulse that has been convertedusing third-order cross-polarized wave generation (XPW). The ASE showsas a pedestal background in the interference pattern. The energy contrastbetween the short-pulse component and the ASE is retrieved. The spectraof the interacting beams are obtained through an improvement to theself-referenced spectral interferometry (SRSI) analysis.

© 2014 Optical Society of America

OCIS codes: (190.7110) Ultrafast nonlinear optics; (190.1900) Diagnostic applications of non-linear optics; (320.5520) Pulse compression; (320.7100) Ultrafast measurements.

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Phys. Rev. Lett. 62, 155–158 (1989).7. B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-

femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749 (1996).8. D. E. Adams, T. A. Planchon, A. Hrin, J. A. Squier, and C. G. Durfee, “Characterization of coupled nonlinear

spatio-spectral phase following an ultrafast self-focusing interaction,” Opt. Lett. 34, 1294–1296 (2009).9. C. Rodel, M. Heyer, M. Behmke, M. Kubel, O. Jackel, W. Ziegler, D. Ehrt, M. Kaluza, and G. Paulus, “High

repetition rate plasma mirror for temporal contrast enhancement of terawatt femtosecond laser pulses by threeorders of magnitude,” Appl. Phys. B (2010).

10. H. J. Liu, Q. B. Sun, J. W. N. Huang, and Z. L. Wang, “Highly efficient pulse cleaner via nonlinear ellipse rotationin liquid CS2 for ultrashort pulses,” Opt. Lett. 38, 1838–1840 (2013).

#211748 - $15.00 USD Received 12 May 2014; revised 4 Jul 2014; accepted 7 Jul 2014; published 17 Jul 2014(C) 2014 OSA 28 July 2014 | Vol. 22, No. 15 | DOI:10.1364/OE.22.017968 | OPTICS EXPRESS 17968

11. A. Jullien, O. Albert, G. Cheriaux, J. Etchepare, S. Kourtev, N. Minkovski, and S. Saltiel, “Nonlinear polarizationrotation of elliptical light in cubic crystals, with application to cross-polarized wave generation,” J. Opt. Soc. Am.B 22, 2635–2641 (2005).

12. A. Ricci, A. Jullien, J.-P. Rousseau, Y. Liu, A. Houard, P. Ramirez, D. Papadopoulos, A. Pellegrina, P. Georges,F. Druon, N. Forget, and R. Lopez-Martens, “Energy-scalable temporal cleaning device for femtosecond laserpulses based on cross-polarized wave generation,” Rev. Sci. Instrum. 84, 043106 (2013).

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1. Introduction

In chirped-pulse amplification (CPA), a technique in common use for modern amplifiers forultrafast pulses, short laser pulses are stretched, amplified and then re-compressed in order toachieve ultra-high peak intensities [1–4]. In addition to the amplified injected pulses, unde-sirable amplified spontaneous emission (ASE) and low-energy satellite pulses emerge at theoutput. ASE is present in all laser amplifiers, since the gain medium integrates and stores thepump energy before it is extracted, but it is particularly an issue in certain architectures suchas high repetition rate regenerative amplifiers and fiber laser amplifiers, where the ASE propa-gates in a common spatial mode with the injected pulse and the gain medium is pumped overa time much longer than the extraction time. Many discussions of ASE are concerned with theintensity contrast between the compressed short-pulse and the background pedestal. In manyhigh-intensity applications, especially with solid-density targets, the ASE and satellite pulsesthat arrive before the main pulse (pre-pulse) can induce nonlinear changes in the target [5]. Forexample, the expanded plasma created by pre-pulse has been found to dramatically enhanceresonance absorption by the main pulse, which can enhance x-ray emission [6] or prevent in-teraction of relativistic intensity pulses with solid density [7]. There are also many situationswhere the energy contrast is the important parameter. If the target linearly absorbs the inci-dent light, the integrated energy in the background can affect the target medium, for example,through heating in metal or semiconductor targets. Even in a simple measurement of a self-phase modulated spectrum, there is a linear sum of the broadened short pulse spectrum and thespectrum of the long pulse. A final example that is relevant to the technique presented in thispaper is our recent interferometric measurement of the nonlinear response of a material [8]. Inthat experiment, the laser beam was split into low and high power arms, with the latter armundergoing a nonlinear interaction with a test material. The beams were then recombined with


a relative delay, so that spatially resolved spectral interferometry showed fringes in the spectraldirection. Fourier analysis of the fringes led to a spatially-resolved measurement of the nonlin-ear spectral phase. In the case of a low energy pedestal, the observed spectral fringes become asuperposition of two interferograms: 1) the ASE interfering with itself, and 2) the desired shortreference pulse interfering with the nonlinear test beam. The superposed interferograms cannotbe separated, and the resulting characterization of the nonlinear phase is compromised.

Even when the intensity contrast is reasonably good (for example 106), the energy contrastmay be poor. Consider, for example a regenerative amplifier system that produces a compressedoutput pulse of 50fs duration. With an output Pockels cell with a 5 ns window to pass the shortpulse and reject the majority of ASE, an intensity contrast of 106 still results in a substantialfraction (> 10%) of energy in the ASE. Several techniques of pulse cleaning that have beendemonstrated, including fast Pockels cells, saturable absorbers, harmonic conversion, plasmamirrors [9] and nonlinear ellipse rotation [10]. One approach that has had recent success iscross-polarized wave generation (XPW) [11, 12], which is a degenerate four-wave mixing pro-cess, producing orthogonally polarized to the input field polarization, high-fidelity pulses. TheXPW output, easily selected with a polarizer, has been used for pulse cleaning in double-CPAsystems [13,14] as well as for hollow fiber pulse compression [15]. Alternatively, optical para-metric amplification can be used as a high-gain, high-contrast front-end to avoid ASE alto-gether [16].

Intensity contrast of laser systems can be measured with third-order autocorrelation [17],where the second harmonic of a pulse is cross correlated with the fundamental to produce thirdharmonic. Because of the wide wavelength separation of the signals, extremely high inten-sity contrast (up to 1012) can be measured (for example [14]). The scan range of the cross-correlation is typically in the sub nanosecond range, not sufficiently long to capture the fulllong pulse pedestal that is at least several ns in duration. A fast photodiode or photoconductivedetector can also be used to estimate the energy before the short pulse, but cannot measure thetotal ASE energy, as it will saturate the moment the main pulse gets detected, or the ASE willbe close to background noise. This method also has limited temporal resolution compared tocross-correlation techniques.

In this paper, we demonstrate the use of XPW and spectral interferometry (SI) to measure theenergy contrast, as well as improve on the already existing techniques to improve the accuracyof measured spectral shape of a laser system output pulse. In one path of an interferometer,the XPW signal is generated, then interfered in a spectrometer with a copy of the input pulsefrom the other path. Since the XPW generation process acts as a temporal filter to pass highintensity, the low intensity background is significantly reduced, and the interference primarilytakes place between the XPW signal and the short-pulse component of the input pulse. Theincoherent ASE shows as a background and results in lower fringe contrast in the interferogram.This allows the determination of the relative energy content of the ASE. The technique makesuse of a similar optical arrangement that is found in self-referenced spectral interferometry(SRSI) [18], in which the XPW signal serves as a reference pulse for the characterization ofthe spectral phase of the input pulse under test. We extend the SRSI retrieval algorithm to findan improved estimate of the shapes of both the fundamental and the XPW spectra. Finally, theretrieved information is used to measure the ASE background spectrum and energy content.

2. Experimental setup and algorithm for analysis

For our experiments, we used a regenerative amplifier of our own design with 1KHz repetitionrate, pumped with 5.8mJ by a frequency-doubled Nd:YLF laser. The amplifier, which has aninternal Pockels cell and second external Pockels cell that is used to select the amplified pulse.The amplifier is seeded with approximately 75pJ per pulse, a small energy relative to other sys-


tems owing to the low oscillator power and the low efficiency of the double-pass grating-basedstretcher. The regenerative amplifier output is capable of producing 400µJ/pulse, 30fs pulsesafter compression. A schematic of the in-line interferometer used to create spectral interfer-ograms is shown in Fig. 1. Although it was not critical, we used an imaging spectrometer inour experiment. The spectrometer had a nominal resolution of 0.134nm/pixel, collected with a1024x1024 TE-cooled CCD array with 16bit resolution. The integration time used for the spec-trometer was 30ms (thirty shots). First a clean-up polarizer (P1) ensures polarization purity ofthe input beam, after which a 1mm thick birefringent calcite plate (CP) is used to produce smalland large amplitude pulses delayed by τ =600fs. Focused by a lens (L1, focal length 400mm),the large amplitude pulse is intense enough to generate an XPW signal in a BaF2 crystal (1mmthick, ¡110¿-cut). After being collimated by another lens (L2, focal length 400mm), it passesthrough the second polarizer (P2) oriented to pass the XPW signal and the low amplitude de-layed pulse from the calcite window. The interference patterns are collected in the spectrometer.The sampling rate in the slit direction was approximately 30µm per pixel. A low percentage(∼2%) of XPW is generated to avoid any saturation and cross-phase modulation effects.

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IF(h)

IA(h)

BaF2Imaging

spectrometerL1 L2 P2CPP1

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Intensity(au)

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IFA(h)

IX(h)

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IINT(h)

Fig. 1. In-line interferometer layout. Polarizer P1 cleans up the input polarization and thecalcite plate CP creates two copies of the input signal with relative delay and oppositepolarization. A 4-f imaging system relays the output of the calcite birefringent plate (CP)to the spectrometer, with XPW conversion of one of the two puIses in the barium fluo-ride crystal (BaF2). Analyzing polarizer P2 selects this XPW signal and also passes thelower energy second pulse. The graphs illustrate the different signals in a simulated datarun for a case where the energy content of the ASE is 25% of that of the fundamentalwave and the ASE central wavelength is slightly lower. IF (λ ), IA(λ ) and IX (λ ) representthe spectral intensities of the fundamental wave, ASE and the XPW signals, respectively.IFA(λ ) represents the combined spectrum of the FW and ASE that would be measured bythe time-integrating spectrometer. The calculated interferogram IINT (λ ) shows a pedestalbackground that results from the presence of the incoherent ASE signal.

For our analysis, we require measurements of the spectral interferogram and the individualspectra of XPW and output of the laser, which contains the short-pulse fundamental wave (FW)and the ASE background. P1 and P2 are first aligned for best extinction without the calcite


or BaF2 in between the polarizers. The calcite plate (CP) is inserted and oriented to maintainextinction, then the XPW crystal is inserted and rotated for best XPW generation. The inputchirp of the pulse is also adjusted for maximum XPW bandwidth. This position allows the col-lection of the individual spectrum of the XPW (I(M)

X ). The nonlinear crystal is then rotated tosuppress conversion and the calcite plate is rotated at a small angle to recover the FW + ASEspectrum (I(M)

FA = IF(ω)+ IA(ω)) which is the output of the laser. Finally, the BaF2 crystal isrotated back to the position for maximum XPW generation. CP is adjusted for best fringe con-trast to collect the interference pattern (I(M)

INT ). As part of the process to calibrate the resolutionof the spectrometer, we also collect an interferogram where the output of the laser is interferedwith itself. In this case, the BaF2 is removed and P2 is rotated to maximize fringe contrast. AMach-Zehnder interferometer could also be used, such as what was used in earlier experimentsthat measured the spatio-temporal dynamics of XPW [19]. Acquiring reference spectra in theMach-Zehnder interferometer is more straightforward, but the inline interferometer is morecompact and, if the optics are not wedged, the parallelism and overlap of the beams is ensured.

Figure 1 also shows simulated spectra for the signals in the measurement. The contributionsto these signals in frequency-space are described by the following expression:

IINT = IF + IA + IX +AF AX exp [i(ωτ +∆φAC)]+AF AX exp [−i(ωτ +∆φAC)] (1)

The interferogram IINT gathered by the spectrometer in Fig. 1 is the sum of the ASE back-ground (non-interfering) and the FW-XPW interference. As shown in Eq. (1), it can be un-derstood as a sum of interfering (IAC(ω)) and non-interfering (IDC(ω)) components. The DC(zero-modulation) component is the sum of the intensities all of the beams (first 3 terms inEq. (1)). The AC component (modulated because of the time delay) has an amplitude that is theproduct of the interfering terms. The relative spectral phase between the two interfering pulsesis also included in the AC term. The AC component is selected with a window in transform(delay) space and re-centered on the grid to place the centroid of the AC peak at zero delay.This removes linear phase terms that are unimportant to the shape of the pulse.

Several approximations and assumptions have been made in writing Eq. (1). First, we assumethat the XPW intensity IX and amplitude AX are solely the XPW signal from the coherentFW pulse and do not have any contribution of the ASE to the XPW signal either from directconversion or mixing with the FW. We also assume perfect contrast for the analyzing polarizerP2 so that there will not be any contribution of FW or ASE leakage to the interference signal.Shot-to-shot variations in the FW pulse and direct interference between the FW and ASE arealso neglected. Finally, by using an imaging spectrometer and selecting signal from the centralportion of the beam, any relative variation of spatial phase difference between the FW andXPW signals will not significantly contribute to a reduction of fringe contrast. A more completediscussion of these issues will be found in Section 3.

The simulation plots in Fig. 1 illustrate the different spectral terms for an example where theASE and FW have different spectral shapes. The ASE background results in reduced contrastin the interferogram. Even though the relative pulse energies are adjusted to optimize fringecontrast, there are other contributions to imperfect contrast, such as resolution limits in thespectrometer, scattered light, and local mismatch of the FW and XPW spectral shapes. Theanalysis described below accounts for these effects.

Our full algorithm, presented in Fig. 2, consists of two loops, one for spectral phase (blue)and the other for spectral amplitude (red). Note that the superscripts M and R denote measuredand retrieved (calculated), signals, respectively. Using a third-order polynomial calibration ofthe spectrometer camera pixels to wavelength, we use third-order interpolation on the measuredspectra and interferogram, then resample on a grid with equal spacing in ω . Since our two-pulse


Initialization:Select a starting guess for FW

A(R)F (ω)←

√IAC(ω)

φ(R)F (ω)←4φ(M)

AC (ω)

Run SRSI phase loop:

For current E(R)F (ω):

get best φ(R)F (ω);

save current E(R)X (ω)

Improve FW spectrum:

A(R)F (ω)← I

(M)AC (ω)/A

(R)X (ω)

Get spectral RMS:

ε =∫|A(R)

AC(ω)− A(M)AC (ω)|dω

εi < εi−1

Spectral results:

{E(R)F (ω);E

(R)X (ω)} found

Calculate XPW:E

(R)F (t) = F−1

{A

(R)F (ω)exp[iφ

(R)F (ω)]

}

φ(R)X (ω) = arg

{F{XPW[E

(R)F (t)]

}}

Improve FW phase:

φ(R)F (ω)←4φ(M)

AC (ω)+φ(R)X (ω)

Get phase RMS

4φ(R)AC(ω) = φ(R)F (ω)− φ(R)X (ω)

ε =∫|4φ(R)AC(ω)−4φ

(M)AC (ω)|dω

εi < εi−1

Loop results:

{φ(R)F (ω);E(R)X (ω)} found

Scale energies of:

A(R)F (ω), A

(R)X (ω), A

(R)AC(ω)

Calculate ASE spectrumand energy:

A(R)A (ω) = A

(M)FA (ω)−A(R)

F (ω)

no

yes

no

yes

Fig. 2. Proposed SRSI algorithm flowchart. The components of the phase and amplituderetrieval loops are outlined in blue and red, respectively. A brief description of each stepis displayed at the top of each box in bold. φ refers to phase, I to intensity, A to realamplitude, E to a complex electro-magnetic field. The superscripts (M) and (R) refer tomeasured and retrieved data, respectively. Subscripts F , A, X , FA correspond to FW, ASE,XPW, and FW+ASE respectively. The function XPW[...] returns the XPW field from thetime-domain FW field. The ε terms represent the root-mean-square (RMS) error, and thesubscript i denotes the iteration index.

delay is sufficiently small to avoid pushing the Nyquist sampling limit, the resampling artifactsdescribed by Albrecht [20] and Dorrer [21] are not expected to greatly decrease the dynamicrange of our measurement. The alternative method of resampling described in those paperscould be employed to avoid interpolation-induced noise. The spectral phase loop generallyfollows the SRSI algorithm [18], except in the initial choice of the spectrum of the fundamental.Instead of using the measured spectrum of the laser, I(M)

FA (ω), which can be distorted by thepresence of ASE, we derive our initial guess for the FW spectrum by the following procedure.We select one of the AC peaks of the Fourier transform of I(M)

INT (ω) with a tapered window. Thespectral window used to select the AC peak extended from halfway between the DC and ACpeaks and was symmetric around the DC peak. This excluded any high frequency contributions,but for our measurements, there was no signal observed above the noise level in that frequencyrange. The AC peak was then shifted to place the centroid at the center of the grid to removeunimportant linear spectral phase terms. The resulting signal was finally transformed back tofrequency space. The amplitude of this signal is the product between the actual XPW and FWfields, so we used its square root as the starting amplitude of the FW (A(R)

F (ω)). The measuredspectral phase difference ∆φ

(M)AC (ω) was used as the initial guess for the FW phase. The phase

of the AC peak represents the measured spectral phase difference between the short pulse andthe XPW,4φ

(M)AC .


The phase-retrieval loop of the original SRSI algorithm is then calculated. Using the initialguess of the time-domain field of the fundamental E(R)

F (t), the corresponding XPW field E(R)X (t)

is calculated. Since a linear spectral phase on the pulses is not important for the algorithm, weposition the centroid of the pulse on the temporal grid as described above. This re-centeringis particularly helpful for pulses with odd orders of phase that are asymmetric in time. Aftertransforming back to ω−space, an improved guess for the spectral phase of the fundamental(φ (R)

F (ω)) can be obtained by adding the retrieved XPW spectral phase to the measured spectralphase difference (4φ

(M)AC ). The updates lead to a new calculated 4φ

(R)AC from which we can

calculate the error ε by integrating the root-mean-square (RMS) difference from4φ(M)AC across

the laser bandwidth. Since the loop can sometimes stagnate with an error that oscillates withiteration, we monitor the average error of several loops and stop the iteration when this averageerror stabilizes. The oscillatory behavior was noticed predominantly when using the the squareroot of I(M)

FA (ω) as the starting FW spectrum as is conventional for the SRSI algorithm. At theend of the phase portion of the algorithm the retrieved FW (E(R)

F (ω)) and XPW (E(R)X (ω)) are

saved and used as input for the spectral part the algorithm.As described above, the spectral content of the measured FW+ASE may be different from

the short-pulse component, for example, if the amplifier is seeded with a pulse that is awayfrom the spectral peak of the gain. One indication that the assumed FW spectrum for the SRSIalgorithm is incorrect can be seen by comparing the XPW spectrum calculated from the SRSIalgorithm with the measured XPW spectrum. Another check is to compare the shapes of themeasured and calculated AC spectrum (AX (ω)AF(ω)). We use the latter spectrum to improveour estimate for the fundamental spectrum since it is less influenced by measurement noise. Theestimate for the FW spectrum to be used for the next iteration of the SRSI loop is improved bydividing the retrieved XPW spectrum into the measured AC peak: I(R)F (ω) = I(M)

AC (ω)/I(R)X (ω).This loop continues until the current product of the retrieved XPW and FW spectra is consistentwith the measured I(M)

AC (ω) using a 5% threshold for the RMS-difference value.As noted in previous work [18], the accuracy of the SRSI phase retrieval algorithm dimin-

ishes with large amounts of even-order phase when the spectrum is smooth and symmetric. Forexample, an input pulse with strong second-order phase will lead to an XPW spectrum that isnarrower than the FW spectrum and with a similar chirp rate. Since the chirp rates of the twosignals are comparable, the measured spectral phase difference is small, and the SRSI algorithmwill retrieve with a phase with a second order phase that is lower than that of the input pulse.Our algorithm shares this sensitivity to input second order phase: if a phase is retrieved with toolittle second-order component, the predicted XPW spectrum will be larger than it should be, andthe new spectral estimate of the FW obtained by dividing the AC peak by the XPW spectrumwill be too narrow. When the FW is retrieved at the end of our algorithm, we check the XPWretrieved spectrum against what was experimentally measured (the two curves are normalizedto have the same area). When the input second-order phase is not large, the RMS differencebetween the measured and retrieved spectra is typically less then 5%. In practice, the constrainton input second-order phase is not a severe limitation, since most amplifier systems include apulse compressor that can be adjusted for maximum XPW bandwidth as discussed above inour alignment procedure. Figure 3(a) shows a comparison of the measured (solid) and retrievedXPW spectra without (red) and with (green) the spectral loop. In this example, where there wasapproximately 50% ASE energy, the RMS difference error on the XPW spectrum without thespectral loop was 22%. Application of the full phase and spectral optimization loops retrievedan improved guess for the XPW spectrum (4.9% RMS error). Figure 3(b) illustrates the capabil-ity of the spectral loop using the model conditions from Figure 1. The retrieved XPW spectrum(green, dashed) lies on top of the spectrum calculated from the input FW pulse (black, solid).


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(a)

q

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(b)

Fig. 3. XPW spectra: measured (solid line), retrieved without spectral loop (green), re-trieved with spectral loop (red). (a) Analysis of experimental data. (b) Example with modelusing conditions from Fig. 1.

The recovered spectral shapes for the FW and XPW improve the confidence in characterizingthe true short pulse spectrum and pulse shape in the presence of ASE background. It is possi-ble to go further, and estimate the energy content and the spectral shape of the FW and ASEcomponents of the laser output. Our modeling shows that if the spectral shapes and energies ofthe FW and XPW are known, these signals can be subtracted from the DC peak of the interfer-ogram to obtain the incoherent background. In real measurements, there are two challenges toobtaining quantitative energy information from the interferograms.

The first of these is to determine the relative energy contribution of the XPW to the interfero-gram. In our measurements, we recorded the spectra of the interferogram, XPW and FW+ASEsequentially. Small fluctuations in the pulse energy and spatial walk off due to pointing varia-tions into the spectrometer can lead to small energy differences of a few percent. In our system,there were no significant changes in spectral shape from shot to shot, so we can correct forthese fluctuations. Noting that the DC peak of the measured interferogram should be the sameas the sum of the measured spectra for the FW+ASE (I(M)

FA (ω)) and XPW (I(M)X (ω)), we can

find a better estimate of the energy balance by weighting the two spectra by factors cFA and cX

I(M)DC (ω) = cFA ∗ I(M)

FA (ω)+ cX ∗ I(M)X (ω), (2)

then adjusting the coefficients independently to minimize the RMS difference ε =∫ {I(M)

DC (ω)−{cFA ∗ I(M)

FA (ω)+cX ∗ I(M)X (ω)}}dω . This ensures that the interference pattern is self-consistent

with the separately measured spectra, even though the image energies required adjustment. Thisprocedure is most robust when there is a significant difference in the shapes of the two spectra,which is typically the case if the pulse is sufficiently well-compressed that the XPW spectrumis wider than the FW. Once these correction factors are known, the measured XPW spectrumcan be subtracted from the DC peak. Subtracting the measured XPW spectrum from the DCpeak will remove any contributions of scattered light from that signal.

Since the phase and spectral retrieval procedure described earlier in this paper led to theshape of the unknown FW spectrum, the second challenge is to determine the energy contentof the FW spectrum. With perfect spectrometer resolution, the FW spectrum with the correctenergy can be found by dividing the square of the AC peak spectrum, i.e. (I(M)

AC (ω))2/I(R)X (ω)→I(R)F (ω). In practice we must account for the line spread function of the spectrometer that leadsto a reduction of fringe contrast that is unrelated to the incoherent background. We determinedthis with two approaches. The simplest approach was to create a spectral interferogram of the


output of the laser with itself, taking care to adjust the energy balance to maximize the fringecontrast. In the Fourier transform of this interferogram, the AC peaks should be one half ofthe height of the DC peak provided that the energies are balanced and that the paths of theinterferometer have equal spectral phase. In this case the test interference pattern probes thespectrometer modulation transfer function (MTF) at one modulation frequency. Any reductionin amplitude of the AC peak we attribute to the MTF of the spectrometer. The influence of anyspectral phase difference between the paths can be avoided by cutting one of the AC peaks fromdelay space, then transforming back to frequency space. The ratio of two times the amplitudeof the resulting spectrum to that of the spectrum from the DC peak gives the correction factorfor the energy of the AC peak in our data.

We also demonstrate a second procedure that is highly accurate but somewhat more laborintensive. The blurring effects from the spectrometer originate from the impulse response func-tion:

Imeasured(ω) = Iactual(ω)⊗h(ω,ω ′), (3)

where Iactual is the un-blurred spectrum, Imeasured is the blurred spectrum, ⊗ represents a con-volution operation, and h is the impulse response function. Deconvolution of the data, as previ-ously demonstrated by Yetzbacher et al. [22], allows us to remove the effects from a system’simpulse response function. To obtain the impulse response of the spectrometer, we measuredthe spectrum from a krypton lamp and finely adjusted the grating in the spectrometer. These ad-justments sweep the peaks of the spectrum across the pixels on the camera. Observing a singlepixel as a peak scans past it gives us a higher resolution measurement of the impulse responsefunction for the spectrometer than the resolution of the camera in the spectrometer. This high-resolution impulse response function gives us the ability to effectively de-blur the measuredinterferogram using the following relation:

Iactual(ω) = F

{Imeasured(t)

H(t,ω ′)

}, (4)

where F represents a Fourier transform operation and tildes represent inverse Fourier trans-formed variables. H is the transfer function, or inverse Fourier transform of the impulse re-sponse function, for the system. In addition to the procedure above, a Gaussian mask was ap-plied to Imeasured to remove high frequency noise contributions that reduce the effectiveness ofthe deconvolution process. This method measures the full transfer function of the spectrometerand has been shown to lead to attosecond phase resolution [22], though the limits in dynamicrange have not been fully analyzed in that work. Our simpler procedure described earlier effec-tively obtains an estimate of the transfer function at a single delay position. The accuracy of thedeconvolution procedure, which is limited by the dynamic range and noise of the detector, setsa limit on the ultimate sensitivity of our technique for measuring the incoherent portion of thebeam.

Finally the ASE spectrum can be calculated from the DC peak I(M)ASE(ω) = I(M)

DC (ω)−(I(R)F (ω)+ I(R)X (ω)). Alternatively, it can also be calculated via the separately collected (andenergy-corrected) FW + ASE spectrum, I(M)

ASE(ω) = I(M)FA (ω)− I(R)F (ω). We have found that both

methods work comparably well when processing with simulated data. The ability to know theintensities of the spectra helps increase the sensitivity of the measurement of background en-ergy to the sub-percent level.

Figure 4 shows data for measurements of the ASE background in our experiment. A lineout of the spectral interferogram is shown in Fig. 4(a). Diminished fringe contrast is seen inthe 760-775nm spectral region of the interferogram. This resulted from scattered light within


750 800 8500

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-400 -200 0 200 4000.0

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750 800 8500

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Fig. 4. (a) Low ASE content (7.5%) interferogram. (b) Transform space representation ofthe interferogram (red) along with measured spectrometer transfer function (dashed). Theblack line results from division by the transfer function. (c) Retrieved FW spectrum (black),low ASE spectrum (solid-red) and high ASE spectrum (dashed-red).

the spectrometer for the broader-band XPW signal that was effectively removed from the ASEcalculation by our procedure described above. Figure 4(b) shows the normalized Fourier trans-form of the interferogram taken of the FW+ASE only (red). The height of the AC peak is seento be lower than the expected value of 1/2, partly because of imperfections of the spectrom-eter resolution. Spectral phase difference in the paths leads to broadening of the AC peak inthis domain. The upper dashed curve in this figure is the measured transfer function of ourspectrometer using the method outlined in Yetzbacher et al [22] described above. Dividing thetransfer function, and selecting the AC peak yields a spectrum that is the same shape as theDC spectrum as expected, and is within a fraction of a percent of one half of its amplitude,confirming the validity of the deconvolution process.

We varied the ASE energy content of our amplifier system by either using or bypassing thePockels cell that followed our regenerative amplifier. Fig. 4(c) shows the retrieved, normal-ized fundamental spectrum and the retrieved ASE spectra with the post-regenerative amplifierPockels cell on (solid) and off (dotted). For low ASE (Pockels cell on) we recover a smoothGaussian ASE spectral shape centered at laser system wavelength. The ASE energy calculatedvia integrating the area under curve of the retrieved spectrum was 7.1% and 7.5% relative tothe FW energy, using the reference interferogram and deconvolution methods, respectively. Forcomparison, the ratio of the energy of the amplifier output unseeded vs seeded was 66%. This ismuch greater than what we measure with our technique because when seeded, much of the en-ergy goes to the injected pulse. When the Pockels cell was turned off and bypassed, the energycontent of the ASE was 131% of the fundamental.

3. Discussion

In the present experiment, we have demonstrated measuring the incoherent content of the am-plifier output at the 7% level, but to extend the measurement sensitivity to levels below thesingle percent level, it is important to consider other sources of degradation of fringe contrast.First, in Eq. (1), any contribution to the XPW signal from the ASE is neglected. There willbe some XPW generated from the ASE, but this contribution will be negligible in most casesowing to the relatively low intensity of this signal. For example, if the FW and ASE signalshave equal energy, but with durations of 100fs and 1ns respectively (10−4 intensity ratio), thecontribution of the ASE to the XPW signal would be 10−12 of that of the FW. More impor-tant would be mixing terms in the nonlinear XPW process between the FW and the ASE. Thestrongest of these terms will be linear in the ASE and second-order in the FW intensities. Theterm EF E∗F EA would have the same phase as the ASE and would therefore produce an inter-


ference pattern superposed on that of the FW-XPW interference. The strength of this artifactinterference pattern would still be lower than the principal one by the ratio of the intensities.More significant is the assumption that the calcite polarizer P2 has perfect contrast, rejecting allof the unconverted input light (FW and ASE) in that path. In practice, we check to see that thecontrast is sufficient by blocking the seed to the amplifier and verifying that the transmission isnear the detection limit of our 16-bit camera at the same exposure time as used to collect thedata. Imperfect contrast will lead to additional interference terms that would add to the desiredmodulation term. The order of magnitude of the resulting distortion of the retrieved spectralamplitude and phase would be approximately equal to the fractional energy leakage.

In principle, the ASE and the XPW signals do interfere, but owing to the slow shutter speedof our camera (30ms), the interference pattern would wash out when averaged over this manyshots. Moreover, even if the data were taken single shot, the large amplitude phase variationsthat cause the ASE pulse to extend over several ns would lead to fringes that would not beresolved by our spectrometer. Mechanical vibrations in the interferometer can lead to degradedfringe contrast, but we find that the inline set up quite stable. As pointed out in a recent pa-per [23], shot-to-shot variations in the pulses can lead to a reduction of fringe contrast thatwould be interpreted by our algorithm as incoherent signal. The presence of these variationscould be detected by performing a frequency-resolved optical gating experiment on the pulse.Finally, we note that if an imaging spectrometer is not used, the fringe contrast can be degradedif there is a difference in spatial phase between the FW and XPW pulses. In our optical ar-rangement, the spectral fringes were observed to be straight, indicating that both beams had thesame divergence. When using an imaging spectrometer, any spatial variation in the fringes isresolved, and in fact the whole image could be used to improve the dynamic range. In futurework we will investigate some of these issues to extend our technique to greater sensitivity.

4. Conclusion

We have presented a technique for measuring the energy and spectrum of the ASE emerg-ing from a pulse amplifier. We achieve this by carefully calibrating the energy content of thepulses making up the interferogram, and adding an improvement to the self-referenced spectralinterferometry (SRSI) algorithm to ensure that the FW and XPW spectra are consistent. Themeasurements described here can be performed with a simple in-line interferometer and a non-imaging spectrometer. We are currently working on new experiment design that can make thesemeasurements in a single shot, which we anticipate will lead to higher accuracy and sensitiv-ity. The analysis is sufficiently fast in computational power that such a measurement could beperformed in real time during the alignment or day-to-day operation of an amplifier system.

Acknowledgments

We acknowledge funding support from AFOSR (FA9550-10-1-0394) and from DOE (DE-SC0001559). CD and AM acknowledge funding support from AFOSR (FA9550-10-1-0561).


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