Compression of high-energy ultrashort laser pulsesthrough an argon-filled tapered planar waveguide

Shihua Chen,1,2 Amelie Jarnac,1 Aurélien Houard,1 Yi Liu,1 Cord L. Arnold,3 Bing Zhou,1

Benjamin Forestier,1 Bernard Prade,1 and André Mysyrowicz1,*1Laboratoire d’Optique Appliquée, École Nationale Supérieure de Techniques Avancées ParisTech, École Polytechnique,

CNRS UMR 7639, Palaiseau 91761, France2Department of Physics, Southeast University, Nanjing 211189, China

3Department of Physics, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden*Corresponding author: andre.mysyrowicz@ensta‐paristech.fr

Received January 4, 2011; revised February 17, 2011; accepted February 21, 2011;posted February 25, 2011 (Doc. ID 140321); published April 5, 2011

We propose a hollow tapered planar waveguide for compression of high-energy ultrashort laser pulses. Directmeasurements suggest that it seems to find a very good trade-off among the energy throughput, the beam focu-sability, and the pulse compressibility. With a Ti:sapphire laser pulse of 12:0mJ and 40 fs, our experiment pro-duces an output pulse of 9:4 fs duration with energy 9:1mJ (transverse magnetic mode) or 10:0mJ (transverseelectric mode) in argon, each exhibiting a nice spatial mode. To evaluate such a tapered waveguide, the linearwave propagation theory and the solution to its complex propagation constant are also presented. © 2011 OpticalSociety of America

OCIS codes: 320.5520, 230.7390, 320.7110.

1. INTRODUCTIONHigh-energy light pulses down to few optical cycles are key toa wide variety of applications, such as optical frequencycombs, supercontinuum sources, and generation of attose-cond coherent radiation, driving a quest for reliable, stable,and cost-efficient lasers and light pulse manipulations [1–3].On the laser side, the necessary technology has matured;e.g., Ti:sapphire amplifiers can easily lift the pulse energyto the millijoule–joule level for a pulse duration longer than20 fs [2]. As such for the ultrafast optics community, one ofthe most interesting challenges concerns how to compressthe laser pulse to a desired duration with a nice, focusablespatial mode and a high-energy throughput.

To date, there are several compression schemes that lead topulse durations down to the few-cycle regime. Traditionally,hollow core fibers [4] and femtosecond filamentation [5] arecommonly used methods for extreme pulse compression be-low 5 fs but with the output energy mostly confined within thesubmillijoule level due to material damage or intensity clamp-ing. For the former scheme, to increase the output energy,there has been significant effort devoted to steering clearof the onset of gas ionization as the pulse experiences self-phase modulation in gas; e.g., Bohman and coworkers realizedgeneration of 5 fs, 5mJ pulses in helium [6] by applying an in-creasing pressure gradient along the hollow fiber [7]. Note-worthily, the optical parametric chirped-pulse amplificationtechnique [8] seems to provide a promising route to the inten-sity scaling problem, but it requires another pump laser and anelaborate setup. By contrast, a recently proposed hollow pla-nar waveguide (PWG) scheme [9,10], regardless of its intrinsicleaky-mode nature [11,12], defines an interesting compromisebetween pulse compression and energy throughput. But thespatial-mode quality is still a critical issue that entails loweringthe gas pressure for pulse compression [13].

In this article, we propose a hollow tapered PWG (tPWG)made flared forward [14] for compression of high-energy ul-trashort laser pulses, primarily motivated by a recent technol-ogy focus that claimed that a tapered waveguide triumphsover the ridge and stripe ones when designed for high-powerdiode lasers [15]. This innovative idea, although less fre-quently mentioned before, indeed has its origin; as early as1965, Vogel used tapered dielectric tubes and rods for trans-mission of high-power laser light [16]. Here we demonstratethat, when applying to high-energy ultrashort pulse compres-sion, such a flared-taper configuration has the merit of com-bining the high output powers with the high beam qualitypreviously not easily attainable in a conventional PWG(cPWG, for later distinction), thereby enabling the pulse dura-tion to be shorter than can be expected.

2. LINEAR WAVE PROPAGATION THEORYAND CALCULATION OF COMPLEXPROPAGATION CONSTANTAs a starting point, we consider linear wave propagation in atPWG as sketched in Fig. 1, which consists of two polishedfused-silica slabs with 2a the gap width at the entrance andwith 2b at the exit. Let r and θ be the polar coordinates inthe plane of incidence and assume there to be no componentof the field varying in the direction perpendicular to the planeof incidence (say, y direction). Therefore, the propagation ofelectromagnetic radiation at the carrier wavelength λ in such awaveguide can be described by the Helmholtz equation [17]:

∇2ψ þ k2ψ ¼ 0; ð1Þ

where ψðr; θÞ is the y component of the electric field or of themagnetic field, corresponding to the transverse electric (TE)mode or transverse magnetic (TM) mode situation, ∇2 is the

Chen et al. Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. B 1009

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Laplace operator in ðr; θÞ, and k ¼ n0k0 is the wave number inmedium with index n0 ¼ ni

0 in the gap and n0 ¼ ne0 in the

slabs (k0 ¼ 2π=λ).We note that, if the field does not penetrate into the con-

fining plate material, i.e., vanishes at the boundary θ ¼ �θ0(θ0 is the tilt angle of the plate), Eq. (1) has exact eigenmodesolutions, given by [18]

ψmðr; θÞ ¼ Hð1Þκ ðni

0k0rÞ cosðκθ − φmÞ; ð2Þ

where Hð1Þκ is the Hankel function of the first kind of (frac-

tional) order κ ¼ ðmþ 1Þπ=2θ0. Here for definiteness, weuse even integer m for even TE/TM modes (φm ¼ 0) andodd m for odd TE/TM modes (φm ¼ π=2) [17]. Under the cir-cumstances, the propagation constant β in the r direction, de-fined through ψmðr; θÞ ¼ Ψðr; θÞ expðiβrÞ [here Ψðr; θÞ is themode amplitude], can be found from the asymptotic expan-sion of Hð1Þ

κ ðni0k0rÞ in Eq. (2). For our present conditions,

we always have ni0k0r > κ ≫ 1 and as a result have an approx-

imation Hð1Þκ ðni

0k0rÞ≃ exp½iκðtanϑ − ϑÞ − iπ=4�=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπκ tanϑ=2

p,

where ϑ ¼ arccosðκ=ni0k0rÞ [19]. Hence, by direct compari-

son, we obtain

βðrÞ≃ ni0k0

�1 −

ðmþ 1Þ2π28ðni

0k0rθ0Þ2�: ð3Þ

In reality, the field in the confining material is nonzero dueto mode leakage [11,12]. Hence the propagation constant βgiven by Eq. (3) should be modified, that is, should be com-plex, so as to reflect the leaky-mode nature. For this end, weuse a modified ray-optics approach [17] and solve for the com-plex propagation constant in a perturbed manner βðrÞ ¼βð0ÞðrÞ þ iδðrÞ=2 [20], where βð0Þ is the phase delay per unitpropagation distance and δ is the perturbed term accountingfor the power loss.

As illustrated in Fig. 1, we assume now that there exists anarbitrary ray of grazing incidence ðαin ≪ 1Þ, which passesthrough the point Pðr; θÞ. The ray trace can thus be deter-mined by r ¼ rin sinðαinÞ= sinðαÞ, where rin is the polar radiusof incident ray associated with αin, and α is the angle formedat P point (see Fig. 1). In order to build up a field mode in thewaveguide, it is required that the phase shift along the θ̂ direc-tion encountered by the original wave must be an integer mul-tiple of 2π radians of that encountered when the wave reflectstwice [17]. Therefore, we can write the resultant modecondition as

4rinθ0ni0k0 sinðαinÞ ¼ 2ðmþ 1Þπ; ð4Þ

where m is an integer defined as in Eq. (3). Notice that eachphase shift introduced by the reflection at the interface, which

has a value of either 0 or 2π [17,20], has been combined intothe term on the right-hand side of Eq. (4). Considering thatβð0ÞðrÞ ¼ ni

0k0 cosðαÞ, it follows readily from the above modecondition, to the first order, that

βð0ÞðrÞ≃ ni0k0

�1 −

ðmþ 1Þ2π28ðni

0k0rθ0Þ2�; ð5Þ

a formula shown to be the same as the one derived in thenonleaky situation; see Eq. (3).

The attenuation coefficient δ at P point can be evaluatedaccording to the definition of power loss, namely,

YJj¼1

jrðjÞF j2 ¼ e−δðrÞðr−rinÞ; ð6Þ

where rðjÞF is the Fresnel’s reflection coefficient at the jth re-flection, and J ≃ αinð1 − rin=rÞ=2θ0 is the total number of re-flection as the ray reaches the point Pðr; θÞ. As one can verify,at the condition of grazing incidence ðαin ≪ 1Þ, the followingapproximation for rðjÞF holds

lnð−rðjÞF Þ≃ −2 sin½αin − ð2j − 1Þθ0�

Δ ; ð7Þ

where Δ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffin2ei − 1

qfor TE modes and Δ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffin2ei − 1

q=n2

ei forTM modes, with nei ¼ ne

0=ni0 > 1. Then, by use of Eqs. (4), (6),

and (7), we find that

δðrÞ≃ ðmþ 1Þ2π24ðni

0k0rθ0Þ2Δaþ rθ0

a2: ð8Þ

As a result, by combining Eqs. (5) and (8) into βðrÞ, thecomplex propagation constant for either TE or TM leakymode follows readily:

βðrÞ≃ ni0k0

�1 −

ðmþ 1Þ2π28ðni

0k0rθ0Þ2�þ iðmþ 1Þ2π28ðni

0k0rθ0Þ2Δaþ rθ0

a2: ð9Þ

One can prove that, if θ0 approaches zero (henceforthrθ0 → a), this leaky-mode constant, Eq. (9), can reduce ex-actly to the one presented in [9–11]. Importantly, the attenua-tion term δðrÞ here provides an estimate of the leaky-modepower loss; e.g., for the mth single mode, the total power lossin units of decibels per meter is 4:34δðb=θ0Þ. It states that thehigh-order modes undergo higher power loss than the funda-mental one and that the smaller the waveguide separation is,the more strongly the modes dissipate [11]. Also, due to a largeΔ, the TE modes are expected to have a higher throughput,about 7%–10% larger than TM modes.

3. EXPERIMENTAL RESULTS ANDDISCUSSIONA. Energy ThroughputWe verify these predicted results experimentally through ourTi:sapphire chirped-pulse amplification system, which deli-vers 12mJ pulse energy at 100Hz with 40 fs duration. Thebeam 1=e2 radius is 6:8mm [10], and initially, the polarizationis parallel to the plane of propagation (TM), which can bechanged to the y direction (TE) by a half-wave plate. The

Fig. 1. (Color online) Schematic showing the ray trace (red line) in aflared-taper configuration consisting of two slightly tilted polishedfused-silica slabs.

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desired waveguide was constructed using two polished fused-silica slabs (21 cm long, 1:9 cm wide, and 0:6 cm thick) sepa-rated by 102 μm thick plastic spacers at the entrance and127 μm at the exit. It was then properly placed in a 1:5m longgas cell with two 0:5mm thick antireflection-coated fused-silica windows. We use one cylindrical mirror of 0:75m focallength to focus the beam tightly into the entrance of the wave-guide and use another one of 1:0m focal length to collimatethe exiting beam. The output energy was measured to be9:1mJ for TM mode and 10:0mJ for TE mode, resulting in anenergy throughput of 76% and 83% respectively, each nearly8% smaller than our analytical predictions based on Eq. (8).We argue that this slight discrepancy is mainly due to the rea-listic imperfect coupling efficiency (<100%) that occurredduring launching the incident beam into the waveguide en-trance. Also, the inherent nonlinear propagation [14,21] maybe thought, to some degree, to have limited the energythroughput, leading to the measurements deviating fromthe linear theory above. However, our recent companion work[13], which provided the detailed discussions on the linear andnonlinear propagation, shows that the waveguide throughputis not a strong function of the gas pressure (thus, the nonli-nearity), provided that the interaction regime is below strongionization, which is certainly true in our conditions. There-fore, a linear theory given here can be thought to be sufficientin order to discuss the throughput.

B. Beam FocusabilityOn the other hand, despite that we ascribe a high-energythroughput to our tapered waveguide design [16], an intri-guing question arises as to whether such a flared-taper con-figuration could at the same time improve the beam qualityas compared to the cPWG with a constant separation,∼ðaþ bÞ. To address this question, it requires that the focusa-

bility of the emerging beam be measured under different gaspressures in the waveguide (here we use argon as the fillinggas) [13]. We performed this experiment by following thesesteps: first, weakening the exiting beam by using the reflec-tions of three fused-silica wedges, then focusing it by a 1mfocal length lens directly onto the chip of a CCD camera (uEyeUI-2210-M) placed movable along the optical axis, and, finally,recording the spot images in different positions close to thegeometrical focus from which postprocessing data wereobtained.

Figure 2 illustrates the second moment (2σ) widths of thebeam (taking TM mode for our present purpose) measured invacuum and at 2 bars of argon, respectively, exhibiting a goodagreement with their hyperbolic fits [22] along either theguided or free dimension of the waveguide. Based on thesehyperbolic fitting data, one can readily obtain their beam pro-pagation factor, M2, an important parameter usually used tocharacterize the beam quality. Table 1 summarizes the results,including those obtained at 1 bar and 1:6bars of argon, as wellas those measured in a 127 μm separated cPWG under other-wise identical conditions (see the right two columns). Here,for convenience of comparison, all these values are normal-ized against the M2 values of the input laser along the corre-sponding directions, which were measured at atmospherepressure to be M2

ıg ¼ 1:89 and M2ıf ¼ 1:34. Here the subscripts

g and f denote the M2 values measured along the guided andfree dimensions, respectively, and the subscript ı is specially

used for the input beam just delivered by our Ti:sapphirelaser. It is clearly seen that the beam quality deteriorates un-avoidably as the pressure grows because of small-scale self-focusing [21], and worse along the free dimension than alongthe guided one. However, compared to the cPWG configura-tion, our tPWG introduces much smaller adverse effect on thebeam quality along each dimension, particularly along theguided dimension. It is especially interesting to note that, forthe tPWG design, despite the increasing gas pressure, thebeam quality along the guided dimension remains almost asgood as in vacuum. Here the ratio, M2

g=M2ıg, below unity in va-

cuum and 1 bar suggests the spatial filtering of the tPWG,which tends to clean the higher-order modes excited in thehigh-power laser. All this is not surprising because a flaredtaper was proved to enable high beam quality and high-powerthroughput simultaneously [15].

Note that the deteriorative process developed in the freedimension can be explained within the context of the general-ized ð3þ 1ÞD nonlinear Schrödinger equation, including thehigher-order dispersion and nonlinearity effects such as third-order dispersion, plasma defocusing, and multiphoton ioniza-tion [21]. One can refer to our recent paper [13] for details, andhere we will not go further on this issue.

C. Pulse DurationAt last, we measured the intensity and phase of the com-pressed pulse (TM mode) by using a single-shot frequency-resolved optical gating (FROG) [23]. The spectral phase wascompensated by two pairs of chirped mirrors (∅45mm) with

Fig. 2. (Color online) Second moment (2σ) widths of the beam nearthe geometrical focus measured in vacuum and at 2 bars for a TMmode along the guided or free dimension, respectively, all fitted byhyperbolic curves.

Table 1. Normalized M2 Values of the Emerging

Beam (TMmode) Measured at Different Pressures of

Argon for Two Types of Waveguides

tPWG cPWG

Pressure (in bars) M2g=M2

ıg M2f =M

2ıf M2

g=M2ıg M2

f =M2ıf

Vacuum 0.87 0.95 1.11 1.061.0 0.95 1.23 1.30 1.381.6 1.02 1.46 1.46 1.642.0 1.05 1.68 — —

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total group-delay dispersion of −400 fs2 (8 bounces). Thethird-order dispersion was optimized by using an acousto-optic programmable filter—Dazzler [13]. The temporal inten-sity and phase are shown in Fig. 3, measured at the center ofthe beam and at 2 bars of argon. The measured spatial modescaled on the millimeter paper (left inset) and the FROG tracetaken by a near-UV high-resolution digital camera (right inset)are also presented there. It is shown that, with a pressure of2 bars, a pulse intensity full width at half-maximum (FWHM)as low as 9:4 fs was obtained without introducing too muchmodulation in the spatial mode. We point out that, for the TEmode, the same amount of compression can be achieved witha pressure of 1:8 bars.

4. CONCLUSIONIn summary, a hollow tapered tPWG was put forward for usein high-energy ultrashort pulse compressions, with resultssuggesting a very good trade-off among the energy through-put, the beam focusability, and the pulse compressibility. Withour Ti:sapphire laser of 12mJ and 40 fs, we obtain an outputpulse of energy up to 10mJ and duration as low as 9:4 fsFWHM with a nice spatial mode. This corresponds to a reli-able terawatt ultrashort tabletop laser source, a necessary toolfor probing the dynamics of plasmas in the relativistic inten-sity regime [24].

ACKNOWLEDGMENTSThisworkwaspartially supportedby theDirectionGénérale del’Armement (DGA) through grant 200795091 (A. Mysyrowicz),the National Natural Science Foundation of China (NSFC)through grant 10874024 (S. Chen), the Qing Lan Project ofJiangsu Province (S. Chen), and the Deutsche Akademie derNaturforscher Leopoldina through grant BMBF-LPD 9901/8-181 (C. L. Arnold).

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Fig. 3. (Color online) Intensity profile and phase of the compressedpulse measured at 2 bars of argon, with the spatial mode and FROGtrace being shown in the left and right insets, respectively.

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