Measurement of the Page function of an ultrashort laser pulse

15 October 2001

Optics Communications 198 (2001) 163±170

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Measurement of the Page function of an ultrashort laser pulse rster, R. Sauerbrey K. Michelmann, U. Wagner, T. Feurer, U. Teubner *, E. Fo Institut f ur Optik und Quantenelektronik, Friedrich-Schiller-Universitat Jena, Max-Wien-Platz 1, 07743 Jena, Germany Received 25 April 2001; received in revised form 17 August 2001; accepted 21 August 2001

Abstract We demonstrate an experimental setup based on an ultrafast optical plasma shutter which measures the Page function of an ultrashort laser pulse. The Page function is a joint time frequency distribution function that allows for a direct reconstruction of the phase and the amplitude of the electric ®eld of an ultrashort optical pulse without employing iterative algorithms. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 42.65. k; 42.30.Rx Keywords: Page function; Femtosecond laser pulse measurements; Ultrafast temporal measurement; Ultrafast optical switch; Ultrafast optical transient phenomena; Multiphoton ionization; Ultraviolet spectra

1. Introduction The technology of ultrashort laser pulse generation has made large progress in the last decades and today's femtosecond laser systems are widely available in many laboratories [1]. Femtosecond laser pulses ranging from near infrared to ultraviolet are commonly used in numerous applications in many ®elds of science. Examples are ultrafast spectroscopy (see, e.g., Refs. [2,3]), quantum control of chemical reactions [4], the study of nonlinear optical phenomena [3] and highintensity laser±matter interaction [5,6], such as high-order harmonic generation [7] and X-ray generation [8]. For many of those applications the temporal and spectral characterization of the fs-laser pulses

*

Corresponding author. Fax: +49-3641-94-72-62. E-mail address: [email protected] (U. Teubner).

is necessary. However, temporal characterization of such short laser pulses is in general a dicult task because direct measurements are limited to the time resolution of streak cameras (presently approximately 500 fs). Hence, indirect methods, such as autocorrelation measurements, are well established [9]. More recently, new methods have been developed that provide additional information especially on the phase evolution and reconstruction of the complete electric ®eld [10,11]. Such methods are frequency resolved optical gating (FROG) [12], spectral phase interferometry for direct electric-®eld reconstruction (SPIDER) [13] or spectral analysis after propagation through a medium with Kerr-like nonlinear response [14]. Here, we report on a new method for the temporal characterization of fs-laser pulses. To the best of our knowledge the present work is the ®rst to report on a measurement of the Page function of an ultrashort optical pulse. In principle, the Page function yields a complete measurement the

0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 5 1 7 - 6

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K. Michelmann et al. / Optics Communications 198 (2001) 163±170

electric ®eld of the laser pulse by direct transformation without using an iterative algorithm. The experiments were performed with a femtosecond UV laser but the method may be extended to pulses in the VUV range.

In the special case when h…t Dt† is the Heaviside step function a close inspection reveals that S…Dt; x† is equivalent to the integral over the Page function of the laser pulse Z S …Dt; x† ˆ

The basic idea of the method is the following. An ultrashort pump pulse is focused to a spot size of several 10±100 lm on a thin transparent optical material, such as glass (for visible or near infrared radiation) or quartz (for UV laser pulses). The pump pulse intensity is sucient to induce a fast transition from high to low transmission of the originally transparent plate. The mechanism of the ultrafast transition will be discussed in Section 3. A weak replica of the pump pulse, the probe pulse, is spatially overlapped with the pump pulse. The intensity is low enough to avoid plasma formation and other nonlinear e€ects, such as self-phase modulation. By changing the temporal delay Dt between both pulses, which are propagating nearly parallel, the transmission of the probe pulse is well controlled and hence the time integral of the probe pulse intensity is measured. Further information, such as central wavelength and the phase modulation of the probe pulse, is obtained by an additional measurement of the probe pulse spectrum behind the plate. If we assume that the change of the transmission is given by the gate function h…t Dt† the spectrum S…Dt; x† as a function of the pump±probe delay Dt is 1 1

dt E…t†h…t

Dt†e

1

dt P …t; x†;

…2†

where the Page function P is de®ned by

2. Principle and optical arrangement

Z S …Dt; x† ˆ

Dt

2 ;

ixt

…1†

where E…t† is the complex electric ®eld describing the probe pulse. The measured spectrum S…Dt; x† is a member of a class of time frequency functions such as the well known Wigner function and Page function [15]. In particular, the Page function [16] is often used and measured for wavefront sensing [17]. Here we show another application, namely in the ultrafast time domain.

Z o P …t; x† ˆ ot

t 1

0

0




dt E t e

2 :

ixt0

…3†

From Eq. (2) it is obvious that the Page function is obtained by calculating the temporal derivative of the measured time frequency distribution P …Dt; x† ˆ

o S …Dt; x†: o Dt

…4†

As will be shown in Section 3 the gate function h…t† has a very fast rise time compared to the probe pulse E…t† and Eq. (4) becomes a good approximation. A direct state reconstruction of E…t† with amplitude and phase from the measured spectrum S…Dt; x† is possible by using the inversion given by Cohen [15]: Z 1 Z 1 Z 1 1 S … x; x† E…t† ˆ dx dx dh 2pE …0† 1 1 1 /…h; t†   t   exp ixt ‡ ih x …5† 2 with /…h; t† ˆ

Z

 h u

t e 2

ihu

 t h u‡ du 2

…6†

(note: the phase is obtained except for an additive constant). An alternative and more robust method to extract the intensity pro®le and the phase modulation from the S…Dt; x† is to calculate the conditional moments (in particular, when there is signi®cant noise in the spectrograms). According to Eq. (1) the zeroth moment is given by integrating the measured signal with respect to the frequency Cohen [15]: Z 1 Z Dt S …Dt; x† dx ˆ I …t† dt: …7† 1

1

K. Michelmann et al. / Optics Communications 198 (2001) 163±170

Similarly, the ®rst moment is given by: Z Dt Z 1 xS …Dt; x† dx ˆ x…t†I …t† dt 1

1

…8†

and together with Eq. (7) it may be used to extract the instantaneous frequency x…t†. For Eqs. (7) and (8) we have assumed that h…t† can be approximated by the Heaviside step function. This is reasonable because as will be shown below (Section 3), the induced transition from a completely transitive to a completely opaque state can be regarded as nearly step like. Further simulations have shown that the procedure works even for a mathematical error function as the gate function with a ®nite transition time sS , as long as sS is less than 20% of the probe pulse length sprobe .

3. Physical process of the switch As discussed in the previous section the method bases on an ultrafast transition from a highly transparent to a highly opaque state in a suciently short time. The transition time sS may be de®ned by the time interval that is necessary to change the transmission from 90% to 10%. Numerical simulations have shown that sS has to be less than 20% of the probe pulse duration sprobe and in principle any physical process that ful®lls this requirement may be employed. The process used for the present work is ultrafast ionization of the surface of a transparent bulk material. In order to calculate the change of transmission caused by the pump pulse we have developed a model which is based on optical ®eld ionization (OFI) [18] and is brie¯y summarized in the following. When the pump pulse with an intensity of 1015 ±1016 W/cm2 impinges on the surface, a fraction of the laser light is absorbed and a thin surface layer is rapidly heated and ionized. Within a few femtoseconds ultrafast OFI occurs, mostly multiphoton and/or tunnel ionization, and a plasma is created at the front surface of the material plate. The surface plasma has a high electron density and thus becomes opaque. Inverse Bremsstrahlung absorption and ionization by collisions (CI) play a minor role at early times and become important

165

only when the originally very low electron density of the insulator has been strongly increased. Using the OFI rates for quartz from Keldysh's model as a function of laser ®eld strength [19] and assuming a Gaussian intensity pro®le of the pump pulse we have calculated the electron density ne as a function of time t. The calculation was performed for a pump pulse with the following parameters: wavelength kL ˆ 248 nm, pulse duration sL ˆ 500 fs (full width at half maximum, FWHM) and peak intensity Ipump ˆ 1015 W/cm2 at t ˆ 0 fs. As may be seen from Fig. 1 the e€ective ionization degree Z  and thus the electron density rapidly increase with time. The electron density ne is normalized to the critical electron density nc ˆ 1:8  1022 cm 3 for 248 nm, i.e. the density where the laser frequency is equal to the local plasma frequency of the surface plasma. In addition to ne …t†, the time evolution of the electron temperature Te was estimated by the kinetic temperature. Although this is only a rough estimate for the evolution of Te …t†, it turns out that an exact calculation of Te …t† is not of critical importance and has nearly no e€ect on the absolute change in transmission and the transition time sS [18]. From these quantities, i.e. ne , Z  , Te , and kL the optical properties of the originally transparent material were calculated as a function of time t. We have used Drude's model [20] together with a

Fig. 1. Estimated electron density (normalized to the critical density; solid line) and e€ective ionization degree (dashed line) as a function of time. For comparison the laser intensity envelope Ipump of the kL ˆ 248 nm, sL ˆ 500 fs pump pulse (FWHM) is shown as well (pointed line). Here Ipump is normalized to its maximum of 1015 W/cm2 at t ˆ 0 fs.

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K. Michelmann et al. / Optics Communications 198 (2001) 163±170

Fig. 2. Calculated total transmission of the quartz plate (T, dashed line). The transmission is changed by the pump pulse (Ipump ; same parameters as in Fig. 1; pointed line). The probe pulse (input Iprobe;in , dot dash line and light gray background; transmitted Iprobe;out , thick solid line and dark gray background) is shown at four di€erent delay times Dt. The probe pulse has the same wavelength and pulse duration but an intensity that is orders of magnitude lower. Iprobe;in …t† and Iprobe;out …t† are normalized to the maximum of Iprobe;in at t ˆ Dt. Ipump …t† is normalized to its maximum at t ˆ 0 fs.

damping factor estimated from the Spitzer theory [21] to calculate the front-side re¯ectivity R, the absorption A, and ®nally the total transmission T of the bulk material. Fig. 2 shows the results including the situation when a probe pulse is introduced which has the same pulse properties with exception of the much lower intensity Iprobe . At early times the plate is nearly fully transparent as expected for solid quartz (Fig. 2(a)) and thus the probe pulse is nearly fully transmitted. Later, ionization becomes signi®cant and the tail of the probe is cut (Fig. 2(b)). Even later, the total transmission decreases from T ˆ 90% to T ˆ 10% within a time interval of sS ˆ 50 fs (Fig. 2(c)). Thus, the plate illuminated by the pump pulse acts as a fast o€-switch and sS is suciently short with respect to sprobe . Finally, only the leading part of the probe pulse is transmitted and shortly before the pump pulse

maximum there is hardly any transmission detectable (Fig. 2(d)). Physically, the ultrafast switching from a highly transparent to a highly opaque state mostly relies on the strong absorption of the probe pulse [18]. The reason for this is the very fast growth of ne and hence the fast increase of the imaginary part of the complex refractive index. The estimated transition time sS for the gate may be regarded as an upper limit because additional CI and breakdown [22] and e€ects such as multiple ionization (see Ref. [23]) have not been included in the present model. 4. Experimental The experimental setup is shown in Fig. 3. In the present work we have used a KrF -laser system

K. Michelmann et al. / Optics Communications 198 (2001) 163±170

167

Fig. 3. Experimental setup.

operating at 1 Hz repetition rate at a wavelength of kL ˆ 248 nm [24]. Depending on the energy of the seed pulse entering the ®nal ampli®er the emerging pulses were subject to self-phase modulation. Thus, by varying the seed energy it was possible to alter the phase modulation of the pulse. The laser energy and pulse duration were 5 mJ and about sL ˆ 500 fs, respectively. The laser beam was split into a strong pump and a weak probe pulse and both pulses were focused at almost normal incidence with the same focusing optics (f =4:5 achromatic lens). The two pulses were delayed with respect to each other with an accuracy of 2 lm, i.e. 6.7 fs. The zero delay position was determined by measuring the spatial interference pattern in the focus between the attenuated pump pulse and the probe pulse. The accuracy of this procedure was 100 fs. The focal spot diameter was 4.8 lm (FWHM of the bell shaped beam pro®le) and the plate was polished quartz with a thickness of 1 mm. The pump intensity on the surface was Ipump ˆ 6  1015 W/cm2 . The intensity of the probe Iprobe was more than 7 orders of magnitude lower. Thus both, plasma formation and nonlinear distortion of the probe pulse were avoided. The quartz plate was mounted on an xy-translation stage in a vacuum chamber (pressure below 2  10 3 mbar) and shifted between consecutive shots so that the incident laser pulses always struck a

fresh target surface. The transmitted probe beam was spectrally analyzed by an imaging Czerny± Turner 0.25 m spectrometer and spectra were recorded as a function of the delay Dt from a spatial section where pump and probe pulse fully overlapped. 5. Results and discussion For the following measurements the laser has been operated in two modes. First, a low energy seed pulse was used producing almost no additional phase modulation. Second, a high energy seed pulse was used leading to self-phase modulation and, therefore, to spectral broadening. The self-phase modulation is mainly introduced by the window material of the ®nal KrF ampli®er. If self-phase modulation is accompanied by an appropriate amount of linear dispersion an almost quadratic phase modulation results, i.e. the instantaneous frequency is linearly increasing with time. Both pulses have been analyzed by a singleshot PG-FROG autocorrelator [25] to serve as a reference for the measurements of the Page function. From the FROG traces the FWHM and the second order phase modulation have been extracted. Due to the poor spatial pro®le, the experimental errors are of the order of 10±20%.

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K. Michelmann et al. / Optics Communications 198 (2001) 163±170

Then, for both cases the ultrafast plasma gate was used to record spectra of the probe pulse as a function of the delay time. Assuming an in®nitely rapid change of the transmission from 100% to zero, the measured spectrum as a function of the delay is given by Eq. (1). Even for the present experiment where sS  sprobe Eq. (1) is a good approximation for the measured time frequency distribution S…Dt; x†. The results of both measurements are shown in Fig. 4. The spectrum at large negative delay times corresponds to the original pulse spectrum. Two characteristics are easily identi®ed. First, the spectrum broadens considerably for higher seed intensities as expected for an increased self-phase modulation. And second, the phase modulation is much more pronounced in the case of the higher intensity.

Calculating the derivative with respect to the delay yields the Page function of both pulses. From the Page function the FWHM of the temporal intensity and the phase modulation have been extracted. Fig. 5 shows the zeroth and the ®rst moment of the Page function which directly yield the pulse duration and the phase modulation. The results are summarized in Table 1. The probe pulse duration sprobe (FWHM) and the quadratic phase modulation (i.e. the linear chirp parameter) are given for the two measurements shown in Figs. 4 and 5. In the case of the low energy seed pulse a good agreement within the experimental error is found. In the case of the high energy seed (5.2 mJ) the increase of the chirp due to self-phase modulation in the laser window can be seen by comparison of the time frequency distribution functions:

Fig. 4. Measured time frequency distribution functions for a seed pulse energy of 1.2 mJ (a) and 5.2 mJ (b), respectively. The change of a spectrum with Dt corresponds to the Page function. As an example, a series of spectra is shown below.

K. Michelmann et al. / Optics Communications 198 (2001) 163±170

169

Fig. 5. Zeroth and ®rst moment of the time frequency distribution functions shown in Fig. 4: seed pulse energy (a) 1.2 mJ, (b) 5.2 mJ. The corresponding electric-®eld parameters derived from the zeroth and ®rst moment, i.e. the pulse duration and the linear chirp, are listed in Table 1. Table 1 Pulse parameters derived from the PG-FROG and Page function measurement, respectively Seed pulse energy (mJ)

PG-FROG

Page

sprobe (fs)

Phase (ps 2 )

sprobe (fs)

Phase (ps 2 )

1.2 5.2

671  120 ±

6:8  0:2 ±

564  22 543  5

8:1  4 12:7  0:5

whereas the pulse duration remains constant, the quadratic phase modulation increases by 50%. 6. Conclusion and outlook It has been demonstrated experimentally that the measurement of the Page function is an alternative way for a direct reconstruction of the amplitude and phase of the electric ®eld of an ultrashort laser pulse. The method requires an ultrafast optical switch with a long lifetime which can be realized by a rapidly created plasma.

The present method which is based on OFI of a transparent medium, is even suitable for probe pulses with a wavelength kprobe as short as 120 nm. The limit is given by the fact that the material has to be originally transparent for kprobe , but is rapidly transferred to an opaque state.

Acknowledgements This work was supported by the Max-Planck Society under contract no. 44188 and the Deutsche Forschungsgemeinschaft under contract no. SA 325/3-2.

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