Measuring pulse-front tilt in ultrashort pulse
laser beams without ambiguity of its sign using
single-shot tilted pulse-front autocorrelator

doi:10.1364/OE.14.013131

Measuring pulse-front tilt in ultrashort pulse laser beams without ambiguity of its sign using single-shot tilted pulse-front autocorrelator

Avnish K. Sharma, Rajesh Kumar Patidar, Manchi Raghuramaiah, Prasad Anant Naik, and Parshotam Dass Gupta »View Author Affiliations

Optics Express, Vol. 14, Issue 26, pp. 13131-13141 (2006)
http://dx.doi.org/10.1364/OE.14.013131

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Abstract

The measurement of pulse-front tilt (PFT) in ultrashort pulse laser beams, without ambiguity of its sign, has been carried out using an in-house built single-shot tilted pulse-front autocorrelator (TPFAC) based on second harmonic generation in non-linear crystal. The experiment is carried out for the beam propagation through a single prism and a pair of prisms. Expressions of PFT angle have been derived to account for beam size, beam propagation effects and for arbitrary incidence of the laser beam. The measured PFT angles were found in accordance with theoretically calculated values. The calculated sensitivity of TPFAC is found to be larger than that offered by the other techniques.

1. Introduction:

There has been a great upsurge of interest in generation and use of ultraintense ultrashort duration laser pulses [1–5

T. Tajima and G. Mourou, “Zettawatt-exawatt lasers and their applications in ultrastrong-field physics,” Phys. Rev. Spl. topics accelerators and beams 5, 031301, 1–9 (2002). [CrossRef]

]. The development of such laser systems necessarily involves one or more dispersive elements such as prisms, gratings, tilted optical windows etc. The dispersive elements, in general, introduce a pulse front tilt (PFT) in the ultrashort pulse laser beams, which is directly proportional to the angular dispersion [6–8

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993). [CrossRef]

]. A perfectly aligned pair of dispersive elements (which does not introduce any angular dispersion) may be used to remove the PFT in a laser beam. However, this would introduce a spatial chirp [9

X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in Ultrafast Optics,” Opt. Commun. 242, 599–604 (2004). [CrossRef]

] in the laser beam at the output. Next, some PFT may be present even in perfectly aligned pairs (which introduce no angular dispersion) due to simultaneous presence of spatial and temporal chirp in a laser pulse [10

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12, 4399–4410 (2004). [CrossRef] [PubMed]

]. Further, propagation of a spatially chirped laser beam (without angular dispersion) in a simple glass slab generates a PFT in the beam. To avoid spatial chirp, a pair of disperser in double pass or two pairs of dispersers in single pass configuration are mostly used in experiments involving ultrashort laser pulses. However, in such cases also, even a slight error in parallelism [11–12

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum. Electron. 30, 1662–1670 (1994). [CrossRef]

] may result in a non-zero angular dispersion, leading to a tilted pulse front of the laser beam.

Ultra intense laser systems involving ultrashort laser pulses and beams of large sizes, are quite prone to spatio-temporal or spatio-spectral distortions (PFT or spatial chirp) arising either due to imperfect alignment of the optical elements or due to simultaneous presence of spatial and temporal chirp. In particular, PFT may cause significant increase of pulse duration at the beam focus, resulting in a decreased intensity. On the other hand, PFT in laser beams can also be useful in several applications such as traveling wave pumping of x-ray lasers [13

J. C. Chanteloup, E. Salmon, C. Sauteret, A. Migus, P. Zeitoun, A. Klisnick, A. Carillon, P. Hubert, D. Ros, P. Nickles, and M. Kalachnikov, “Pulse-front control of 15-TW pulses with a tilted compressor, and application to the subpicosecond traveling-wave pumping of a soft-x-ray laser,” J. Opt. Soc. Am. B 17, 151–157 (2000). [CrossRef]

] and dye lasers [7

Z. Bor and B. Racz, “Group-velocity dispersion in prisms and its application to pulse-compression and traveling-wave excitation,” Opt. Commun. 54, 165–170 (1985). [CrossRef]

], optical parametric amplifiers [13–15

], efficient harmonic generation [16

T. Kanai, X. Zhou, T. Liu, A. Kosuge, T. Sekikawa, and S. Watanabe, “Generation of terawatt 10-fs blue pulses by compensation for pulse-front distortion in broadband frequency doubling,” Opt. Lett. 29, 2929–2931 (2004). [CrossRef]

], pulse compression [17

A. Dubietis, G. Valiulis, G. TamoSauskas, R. Danielius, and A. Piskarskas, “Nonlinear pulse compression in the ultraviolet,” Opt. Commun. 141, 50–55 (1997).

], THz generation [18–19

J. Hebling, G. Almasi, I. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large area THz-pulse generation,” Opt. Express 10, 1161–1166 (2002). [PubMed]

] etc. For such applications, the detection and measurement of PFT is essential. This has led to development of several diagnostic tools for spatio-temporal pulse characterization. These include interferometric field autocorrelation [20–21

C. Radzewicz, M. J. la Grone, and J. S. Krasinski, “Interferometric measurement of femtosecond pulse distortion by lenses,” Opt. Commun. 126, 185–190 (1996). [CrossRef]

], spectrally resolved interferometry (SRI) [22

K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, “High precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B 74, S259–S263 (2002). [CrossRef]

], grating-eliminated no-nonsense observation of ultrafast incident laser light E fields (GRENOUILLE) [23

S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE,” Opt. Express 11, 491–501 (2003). [CrossRef] [PubMed]

], spectral phase interferometry for direct electrical field reconstruction (SPIDER) [24

C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B 74, 209–219 (2002). [CrossRef]

] and multiple [15

P. Di Trapani, A. Andreoni, C. Solcia, G. P. Banfi, R. Danielius, A. Piskarskas, and P. Foggi, “Powerful sub-100-fs pulses broadly tunable in the visible from a blue-pumped parametric generator and amplifier,” J. Opt. Soc. Am. B 14, 1245-(1997). [CrossRef]

] or single-shot tilted-pulse front autocorrelator (TPFAC) [25–27

Z. Sacks, G. Mourou, and R. Danielius, “Adjusting pulse front tilt and pulse duration by use of a single-shot autocorrelator,” Opt. Lett. 26, 462–464 (2001). [CrossRef]

]. These instruments vary in the complexity of their setup, information retrieval, parameter regimes, and cost. For instance, while the maintenance-free GRENOUILLE technique cannot be used for laser pulses shorter than 30fs, the interferometry-based instruments are prone to misalignments and cannot be applied to measure PFT in absence of the angular dispersion. On the other hand, the simple setup of TPFAC allows detection, quantitative measurement and optimization of PFT in a visually straightforward manner. However, it is stated in the literature that a TPFAC cannot detect the sign of PFT [23

S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE,” Opt. Express 11, 491–501 (2003). [CrossRef] [PubMed]

] and no report is found on use of TPFAC for such measurements.

In this paper, we demonstrate the use of a single shot TPFAC to detect and measure the PFT without ambiguity in its sign, using a single prism and a pair of parallel prisms, generating PFT in femtosecond laser beams, with and without angular dispersion respectively. A generalized analytical expression for the PFT introduced by a prism is derived along with its dependence on propagation distance. The variation of the PFT with the propagation distance has also been studied experimentally using a 200fs laser beam from an Nd:glass oscillator and a home-built non-linear crystal based TPFAC. This can be useful in either introducing a known PFT or compensating it in ultrashort pulse laser beams required for various applications.

2. Theoretical background

The pulse-front tilt (PFT) in ultrashort laser beams has been well studied both for plane and Gaussian waves [28–32

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986). [CrossRef]

]. The angular dispersion (AD) and hence the PFT of a given beam may change with the propagation distance between the beam waist and the disperser or between the disperser and the observer [28–30

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986). [CrossRef]

]. A model for PFT variation with beam propagation was presented by Martinez [28–29

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986). [CrossRef]

], which was experimentally verified by Varju et al [30

K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27, 2034–2036 (2002). [CrossRef]

] by measuring the AD rather than the PFT. Akturk et al [10

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12, 4399–4410 (2004). [CrossRef] [PubMed]

] extended this model to a temporally and spatially chirped laser beam propagating through a prism kept at minimum deviation. It was theoretically shown (and experimentally confirmed) that the PFT can be present in a beam due to the combined effect of temporal and spatial chirp, even in absence of AD [10

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12, 4399–4410 (2004). [CrossRef] [PubMed]

]. Following the procedure given in Ref. [28

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986). [CrossRef]

] and Ref. [10

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12, 4399–4410 (2004). [CrossRef] [PubMed]

], we derive a general expression for the PFT occurring in a laser beam incident on a single disperser or a pair of dispersers at an arbitrary angle of incidence and for arbitrary distances of incident beam waist and observation point relative to the disperser.

Consider an electric field in the x-ω domain E(x, y, z, ω) of a pulse with a spatial chirp ξ₀ incident on a disperser, as shown in Fig. 1. As the angular dispersion and hence the PFT is assumed in x–z plane, the variable y is ignored from the electrical field description. The field amplitude just after the disperser can be expressed as [28

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986). [CrossRef]

]

E (x, z = 0, ω) = E (α x, ω) \exp (- i k β ω x)

(1)

where k(=2π/λ; λ is the central wavelength) is propagation constant and the parameters α, β are related as

Δ θ = α Δ η + β ω; where α = {\frac{d θ}{d η} ∣}_{ω_{o}} and β = {\frac{d θ}{d ω} ∣}_{η}

(2)

The parameter Δθ in the Eq. (2) is the beam divergence along dispersion direction due to the disperser, η and θ are the angle of incidence and diffraction for the central frequency ν_o, and ω is the frequency shift defined as 2π(ν_o–ν). The Eq. (2) is valid to the first order and is applicable to many experimental situations. The higher order terms in Δφ_i and ω may result in non-linear PFT in laser beams [33

Y. Nabekawa and K. Midorikawa, “High-order pulse front tilt caused by high-order angular dispersion,” Opt. Express 11, 3365–3376 (2003). [CrossRef] [PubMed]

] but are negligible for collimated and small spectral bandwidth laser beams. For plane waves, the parameters α and β can be calculated using dispersion equation for a given disperser. For the case of a prism, these parameters can be given as

α = - \frac{\cos η \cos θ'}{\cos η' \cos θ}; and β = \frac{\sin A}{\cos η' \cos θ} (\frac{d n}{d ω})

(3)

where η′ and θ′ are angles inside the prism at its entrance and exit faces respectively, and A is the prism apex angle (as shown in Fig. 1). For a prism kept at minimum deviation position or for a gratings at Littrow angle position, the parameter α equal to 1. The value of the angular dispersion (β′) for Gaussian beam may be calculated taking into to account the distance (d) of disperser relative to position of incident beam waist, and the distance (z) between the observation point and the disperser, as defined in Fig. 1. The angular dispersion for this case can be given as [30

K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27, 2034–2036 (2002). [CrossRef]

]

β' = β \frac{(1 + \frac{α^{2} z}{d}) + {(\frac{z_{R}}{d})}^{2}}{{(1 + \frac{α^{2} z}{d})}^{2} + {(\frac{z_{R}}{d})}^{2}}

(4)

where z_R is the Rayleigh length, defined as πσ²/λ, σ being the beam waist radius. For a well collimated laser beam (i.e. z_R≫d, α²z≪z_R), β′ becomes equal to β.

Fig. 1. Diagram illustrating the various parameters used in an experiment using a single prism, for a plane wave. In the case of a beam with a curved wavefront, Δθ′≠Δθ [30

K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27, 2034–2036 (2002). [CrossRef]

]

The field amplitude after a distance z from the disperser can be determined by using the Fresnel-Kirchoff”s integral formula as

E (x, y, z, ω) = \frac{i}{λ z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E (x', y', z = 0, ω) \exp [- \frac{i π}{λ z} {{(x - x')}^{2} - {(y - y')}^{2}}] d x' d y'

(5)

Since the dispersion is considered to be in the x–z plane, the variable y in Eq. (5) is ignored throughout the paper and suitable set of constants {b_n} have been used to take care of energy loss, dimensionality and constant phase terms, as was done in Ref. 28. Let the incident laser beam be of the form

E (x, ω) = E (ω) \exp {- \frac{{i k (x - ξ_{0} ω)}^{2}}{2 q (d)}}

(6)

where q(z)=z+iπσ ²/λ, d is the distance of the beam waist from the disperser and E(ω) is amplitude of the laser field in the frequency domain, which can be written in the form

E (ω) = E_{0} \exp {- \frac{{ω^{2} τ_{0}}^{2}}{4}} \exp (- \frac{i φ_{2} ω^{2}}{2}}

(7)

where τ₀ is the FWHM pulse duration of the laser pulse and φ₂ is the quadratic phase (which defines linear chirp). One can solve Eq. (5) as,

E (x, z, ω) = b_{1} \sqrt{\frac{i k q (d)}{2 π z (d + α^{2} z)}} E (ω) \exp [\frac{i k z q (d)}{2 q (d + α^{2} z)} {- β ω + \frac{(x - \frac{ξ_{0} ω}{α})}{z}}^{2}]

\exp [- i k {\frac{β ξ_{0} ω^{2}}{α} + \frac{{(x - \frac{ξ_{0} ω}{α})}^{2}}{2 z}}]

(8)

where b₁ is a constant multiplier. In order to get values of the effective angular dispersion, and the temporal and the spatial chirp from Eq. (8), the following rearrangement of the terms is carried out.

\frac{q (d)}{q (d + α^{2} z)} = \frac{d + \frac{i π σ^{2}}{λ}}{d + α^{2} z + \frac{i π σ^{2}}{λ}} = a_{1} (1 + i \frac{2 α^{2} z}{a_{2} k σ^{2}})

(9)

where a_{1} = \frac{(1 + \frac{α^{2} z}{d}) + {(\frac{z_{R}}{d})}^{2}}{{(1 + \frac{α^{2} z}{d})}^{2} + {(\frac{z_{R}}{d})}^{2}} and a_{2} = 1 + \frac{d^{2}}{z_{R}^{2}} + \frac{α^{2} z d}{z_{R}^{2}}

(10)

Eq. (8) can now be expressed as

E (x, z, ω) = b_{2} c_{1} E_{0} \sqrt{\frac{i k a_{1}}{2 π z} (1 + i \frac{2 α^{2} z}{a_{2} k σ^{2}})} \exp (- \frac{ω^{2} τ_{o}^{2}}{4}) \exp (- \frac{i ω^{2} φ_{2} (z)}{2})

\exp {- \frac{α^{2} a_{1}}{σ^{2} a_{2}} {(x - ξ (z) ω)}^{2}} \exp (- i k β " ω x)

(11)

where b₂ is a constant multiplier and

where b_{2} is a constant multiplier and c_{1} = \exp (\frac{i k (a_{1} - 1) x^{2}}{2 z})

(12)

β " = a_{1} β + \frac{ξ_{o} (a_{1} - 1)}{α z}

(13)

φ_{2} (z) = φ_{2} - a_{1} k β^{2} z - \frac{2 k β ξ_{o} (a_{1} - 1)}{α} - \frac{k ξ_{o}^{2} (a_{1} - 1)}{α^{2} z}

(14)

ξ (z) = \frac{ξ_{0}}{α} + β z;

(15)

Eq. (13) represents the effective angular dispersion (β″) for an initially spatially chirped Guassian beam incident on a prism at an arbitrary incidence angle. For a laser beam without spatial chirp, the effective angular dispersion (β″) for a Gaussian beam becomes a₁β, as expressed by Eq. (4) [30

K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27, 2034–2036 (2002). [CrossRef]

]. For well collimated beams, β″ equals β. Next, Eq. (14) and Eq. (15) describe the position dependent frequency chirp and the spatial chirp respectively, for an arbitrary incidence beam. The PFT can be obtained by first converting the x-ω domain field of laser beam in to x-t domain using inverse Fourier transform of E(x, z, ω) given by Eq. (11) and then locating the peak intensity contour in E(x, z, t), as was done in Ref. 10

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12, 4399–4410 (2004). [CrossRef] [PubMed]

. The time domain field E(x, z, t) can be expressed as

E (x, z, t) = b_{3} c_{1} E_{0} \sqrt{\frac{i k a_{1}}{2 π z} (1 + i \frac{2 α^{2} z}{a_{2} k σ^{2}})} \sqrt{\frac{4 π}{τ'^{2} + i 2 φ_{2} (z)}} \exp (- \frac{x^{2}}{σ'^{2}}) \exp {- {(\frac{t - t_{o}}{τ'})}^{2}}

\exp (- \frac{i φ^{2} (z) υ^{2} (z) x^{2}}{2}) \exp {i (ϕ_{1} (t - t_{0}) + \frac{ϕ_{2} {(t - t_{0})}^{2}}{2})}

(16)

where b₃ is a constant multiplier. The other parameters are as follows

t_{0} = (k β " + φ_{2} (z) ν (z) x

(17)

ϕ_{1} = ν (z) x; ν (z) = \frac{ξ (z)}{ξ^{2} (z) + \frac{{a_{2} σ^{2} τ_{0}}^{2}}{4 a_{1} α^{2}}}

(18)

ϕ_{2} = \frac{φ_{2} (z)}{\frac{τ'^{4}}{4} + {φ_{2}}^{2} (z)}; τ'^{2} = {τ_{0}}^{2} + \frac{4 a_{1} α^{2} ξ^{2} (z)}{a_{2} σ^{2}}

(19)

\frac{1}{σ'^{2}} = \frac{a_{1} α^{2}}{a_{2} σ^{2}} - \frac{υ^{2} (z) {τ_{0}}^{2}}{4}

(20)

The parameter t₀ defines the pulse front arrival time and is used to calculate PFT angle as [10

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12, 4399–4410 (2004). [CrossRef] [PubMed]

]

\tan γ = k β " c + c φ_{2} (z) ν (z)

(21)

The above equation describes the PFT angle for an arbitrary angle of incidence and its evolution with the beam propagation distance z from the disperser. This equation can be easily transformed to the specific cases of beam with no initial spatial chirp [28

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986). [CrossRef]

] and of spatially chirped beam [10

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12, 4399–4410 (2004). [CrossRef] [PubMed]

] comprising of temporally chirped laser pulses. For instance, PFT angle corresponding to an ultrashort pulse laser beams in the limit of negligible spatial chirp (ξ_o/α≪βz) and temporal chirp (φ₂≪ka₁β²z), can be given as

\tan γ = k β c a_{1} \frac{{\frac{a_{2}}{a_{1}} (\frac{σ τ_{o}}{2 α β z})}^{2}}{[1 + \frac{a_{2}}{a_{1}} {(\frac{σ τ_{o}}{2 α β z})}^{2}]}

(22)

The above equation in limit of beam collimation reduces to

\tan γ = \frac{{k β c (\frac{σ τ_{o}}{2 α β z})}^{2}}{[1 + {(\frac{σ τ_{o}}{2 α β z})}^{2}]}

(23)

In the case of a pair of parallel dispersers, a general expression for PFT may be calculated in similar way as that used for the case of single disperser. This is done by first calculating the space-frequency domain field just after the second parallel disperser [Eq. (1)], having dispersion parameter as α′=1/α and β′=-β/α [28

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986). [CrossRef]

], and then transforming the x-ω domain field of laser pulse into x-t domain. Using this procedure, we calculated the PFT angle γ′ after a pair of dispersers (in the limit of negligible spatial chirp and temporal chirp) as

\tan γ' = k \frac{β (a_{1} - 1)}{α} c - \frac{c k a_{1} β}{α [1 + {\frac{a_{2}}{a_{1}} (\frac{σ τ_{o}}{2 α β z})}^{2}]}

(24)

For the case of a well-collimated laser beam (a₁~1 and a₂~1), the above equation reduces to

\tan γ' = \frac{- k β c}{α [1 + {(\frac{σ τ_{o}}{2 α β z})}^{2}]}

(25)

It is clear from Eqs. (25) that in the limit z≫στ_o/2αβ, which is basically the distance where the lateral spectral walk-off exceeds the beam size, the pulse-front tilt angle is equal to the initial pulse front tilt angle due to the first disperser but opposite in sign [compare Eqs. (23) and (25)]. For small distances (z≪στ_o/2αβ) between the dispersers, the PFT may become zero.

We now present an experimental study of the positive and negative PFT due to single prism and double prism respectively using a non-linear crystal based single-shot tilted pulse-front autocorrelator (TPFAC).

3. Description of the single-shot TPFAC

A schematic drawing and a photograph of the experimental setup of the single shot tilted pulse-front autocorrelator are shown in Fig. 2(a) and Fig. 2(b) respectively. The incident ultrashort pulse laser beam was split into two equal intensity beams using a 50% reflectivity beam splitter (50 mm diameter, 6mm thick). One of the beams was horizontally retro-reflected, and the other vertically retro-reflected. The horizontal retro-reflector was also mounted on a precision translational stage to introduce a variable time delay to temporally coincide the two pulses inside the crystal. The laser beam reflected by the vertical retro mirror was once again reflected by a 100% reflecting mirror (50mm diameter) to direct it towards the KDP crystal. The two beams are then made to overlap in the vertical plane in a type I phase matched KDP crystal (20mm×20mm and 6 mm thickness) at a small cross-over angle (2ψ). The spatial distribution of the second harmonic radiation was recorded with a CCD camera-frame grabber-PC combination.

For a tilted pulse-front beam, the autocorrelation trace was a rotated ellipse with its major axis at angle α relative to horizontal direction. For known beam cross-over angle (2ψ) and measured rotation angle (δ) one can calculate pulse-front tilt (γ) as [26

M. Raghuramaiah, A. K. Sharma, P. A. Naik, and P. D. Gupta, “Simultaneous measurement of pulse front tilt and pulse duration of a femtosecond laser beam,” Opt. Commun. 223, 163–168 (2003). [CrossRef]

]

γ = \tan^{- 1} [{1 - (\frac{λ}{n}) (\frac{d n}{d λ})} \tan δ \sin ψ]

(26)

Fig. 2. Experimental setup of TPFAC: a) schematic drawing and b) pictorial view.

The sensitivity of the single shot TPFAC can be obtained from Eq. (26). The variation of the AC trace rotation angle (δ) and its derivatives w. r. to a given PFT angle (γ) for different value of cross-over angle (2ψ) is given in Fig. 3(a) and Fig. 3(b) respectively. It may be seen from these figures that the sensitivity is high for smaller cross-over angles and for small values of the pulse front tilt. For large amount of PFT angles, the rotation of the AC trace saturates [Fig. 3(a)]. This in turn reduces the sensitivity (dδ/dγ) of the instrument [Fig. 3 (b)].

Fig. 3. AC trace rotation angle (a) and its derivative (b) w. r. to the PFT angle at different cross over angles.

4. Measurement of PFT without ambiguity in its sign

Measurements were performed using a diode pumped, CW mode-locked, Nd:glass laser (GLX-200) providing transform-limited 200 fs (FWHM) laser pulses at 100 MHz at a central wavelength of 1055 nm. In single prism experiment, the beam was incident at an angle η=69° on a SF10 prism of apex angle A=60°. The PFT was measured using the above-described TPFAC at different propagation distances from the prism. A typical autocorrelation (AC) trace recorded at a propagation distance (z) of 1m is shown in Fig. 4(a). Figure 4(b) depicts an AC trace corresponding to a laser pulse directly from the laser oscillator. It is seen from Fig. 4(a) that the AC trace lies in 1^st and 3^rd quadrant of Cartesian coordinate system. Some irregularities in the intensity profile of AC traces are due to the local surface condition of the KDP crystal used. A PFT angle of 1.9° was estimated from measured rotation angle (39°) of the AC trace and known beam overlap angle (2ψ) of 12°. Next, it may be mentioned that the observed AC traces also contain the information of pulse duration, which can be obtained simultaneously with the PFT.

Fig. 4. Typical AC trace (a) after prism insertion; and (b) without prism insertion in the 200fs laser beam directly from the GLX 200 laser oscillator.

Similarly, PFT angle was measured to be 1.5° and 0.5° at distance z of 1.9 m and 3.6 m respectively from the prism. These measured values of PFT angle w. r. to the distance z are shown in Fig. 5. The AC trace corresponding to z=1m and z=3.6 m are also shown in this figure. The measured values of the PFT angles are in accordance with the theoretical values of the PFT angle calculated using Eq. (22) for a pulse duration of 200fs, beam waist diameter of 2.4 mm and its distance (d) from the prism of 2m, and negligible initial spatial and temporal chirp.

Fig. 5. The measured and theoretical values of PFT angle as a function of distance (z) from the SF10 prism. (AC trace corresponding to z=1m and z=3.6m are also depicted in the figure)

The PFT angle was also measured when a prism pair was introduced in the laser beam. In this case, the PFT may be present if the lateral spectral walk-off ff (l(dθ/dω)Δω) exceeds the beam size. This can happen when the separation l between the two prisms is kept to be quite large. A PFT of opposite sign, as shown in Fig. 6(a), to that due to single prism was observed for a prism separation l of 3.6m. It is clear to see that AC trace now occupies the 2^nd and 4^th quadrant. This result confirms that the sign of PFT can be measured using TPFAC. No PFT was observed for a short prism separation l of 13 cm, as expected [The AC trace corresponding to this case is given in Fig. 6(b)]. The change in sign of PFT in a prism pair is obvious and is in accordance with Eq. (24).

Fig. 6. Typical AC trace at output of prism pair with a separation of (a) 3.6 m; and (b) 15 cm.

In order to further confirm the sign of PFT, we measured PFT in ultrashort pulse laser beam propagating through a single prism [at an angle of incidence 69° as that the case of Fig. 4(a)] but the laser beam from the prism was now reflected by a 100% reflectivity mirror (in order to spatially invert the PFT) and then made incident on the TPFAC setup. The recorded AC trace in this case is shown in Fig. 7. From this figure, it is clear that the AC trace now occupies the 2^nd and 4^th quadrant in contrast to that shown in Fig. 4(a) for the same magnitude of the PFT.

Fig. 7. Typical AC trace corresponding to PFT angle of opposite sign due to mirror reflection compared to that shown in Fig. 4(a).

5. On the sensitivity of the PFT measurements using TPFAC

Now, we show experimentally that by reducing the beam cross-over angle, one can enhance the sensitivity of PFT angle measurement. Figure 8(a) and Fig. 8(b) depict the AC traces corresponding to same amount of PFT at the beam crossover angle of 12° and 6° respectively. On may see that AC trace rotation angle has increased by a factor of 2 and is in accordance with Eq. (26) as the cross-over angle has decreased by a factor of 2.

Fig. 8. Experimental AC trace for a tilted pulse-front laser beam at beam cross-over angle of (a) 12°; and (b) 6°.

For a given beam cross-over angle, the sensitivity of the PFT measurement is governed by the minimum detectable rotation angle of the AC trace, which depends on the spatial resolution of the CCD camera used to record the AC trace. The accuracy of the PFT angle (γ) measurement using a TPFAC may be determined for a given beam cross-over angle (2ψ) and a CCD camera with 512×512 pixels. In our case, it is estimated to be 24 µrad and 72 µrad at cross-over angle of 12° and 6° respectively and considering accuracy in the vertical and horizontal direction of 8.4µm (1 pixel) and 6.04 mm (512 pixels). The sensitivity of the PFT measurement corresponding to these crossover angles is calculated to be 0.0014 fs/mm and 0.0042 fs/mm respectively. These values are smaller than the one in the GRENOUILLE (0.05 fs/mm [23

S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE,” Opt. Express 11, 491–501 (2003). [CrossRef] [PubMed]

]), which means that the TPFAC may be more sensitive at lower beam cross-over angle. However, in practice, because of the non-uniformity of the laser beam and the nonlinear crystal, one may have large error in measuring the rotation angle of the AC trace. Such errors may be reduced by using a computer program, which finds peak intensity points across the trace and then calculates the rotation angle of the AC trace. Such a computer program was developed using scientific computing software MATLAB. Next, as a TPFAC does not measure the angular dispersion directly, one cannot say whether the PFT is due to any residual angular dispersion or due to simultaneous presence of temporal and spatial chirp. However, in certain experimental geometries such as single disperser, where the PFT is due to angular dispersion, the sensitivity in terms of angular dispersion dθ/dλ, can be calculated and it was estimated to be 0.023 µrad/nm at 1054 nm, which is larger than that obtained in spectrally resolved interferometry (0.2 µrad/nm [22

K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, “High precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B 74, S259–S263 (2002). [CrossRef]

, 31

K. Osvay, A. P. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10, 213–220 (2004). [CrossRef]

]).

6. Conclusion

In conclusion, pulse front tilt measurements in ultrashort pulse laser beams have been performed using a KDP crystal based single shot tilted pulse-front autocorrelator. It is shown that TPFAC can determine the magnitude and sign of PFT in laser beams in a visually straightforward manner with sensitivity higher than that obtained in the other techniques. The TPFAC has been used to measure the PFT angles introduced by single and double prisms. The effect of beam propagation distances has been studied. General expressions for PFT have been derived to account for arbitrary angle of incidence of laser beams with or without angular dispersion.

References and links

1.	T. Tajima and G. Mourou, “Zettawatt-exawatt lasers and their applications in ultrastrong-field physics,” Phys. Rev. Spl. topics accelerators and beams 5, 031301, 1–9 (2002). [CrossRef]
2.	J. H. Sung, J. Y. Park, T. Imran, Y. S. Lee, and C. H. Nam, “Generation of 0.2-TW5.5-fs optical pulses at 1 kHz using a differentially pumped hollow-fiber chirped-mirror compressor,” Appl. Phys. B Laser & Optics 82, 5–8 (2006). [CrossRef]
3.	Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, “Prepulse-free petawatt laser for a fast ignitor,” IEEE J. Quantum. Electron. 40, 281–293 (2004). [CrossRef]
4.	C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004). [CrossRef]
5.	M. Aoyama, K. Yamakawa, Y. Akahane, J. Ma, N. Inoue, H. Ueda, and H. Kiriyama, “0.85-PW, 33-fs Ti:sapphire laser,” Opt. Lett. 28, 1594–1596 (2003). [CrossRef] [PubMed]
6.	Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993). [CrossRef]
7.	Z. Bor and B. Racz, “Group-velocity dispersion in prisms and its application to pulse-compression and traveling-wave excitation,” Opt. Commun. 54, 165–170 (1985). [CrossRef]
8.	J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Opt. Quantum. Electron. 28, 1759–1763 (1996). [CrossRef]
9.	X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in Ultrafast Optics,” Opt. Commun. 242, 599–604 (2004). [CrossRef]
10.	S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12, 4399–4410 (2004). [CrossRef] [PubMed]
11.	C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum. Electron. 30, 1662–1670 (1994). [CrossRef]
12.	K. Osvay, A. P. Kovacs, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, “Measurement of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser,” Opt. Commun. 248, 201–209 (2005). [CrossRef]
13.	J. C. Chanteloup, E. Salmon, C. Sauteret, A. Migus, P. Zeitoun, A. Klisnick, A. Carillon, P. Hubert, D. Ros, P. Nickles, and M. Kalachnikov, “Pulse-front control of 15-TW pulses with a tilted compressor, and application to the subpicosecond traveling-wave pumping of a soft-x-ray laser,” J. Opt. Soc. Am. B 17, 151–157 (2000). [CrossRef]
14.	A. Shirakawa, I. Sakane, and T. Kobayashi, “Pulse-front-matched optical parametric amplificationfor sub-10-fs pulse generation tunable in the visible and near infrared,” Opt. Lett. 23, 1292–1294 (1998). [CrossRef]
15.	P. Di Trapani, A. Andreoni, C. Solcia, G. P. Banfi, R. Danielius, A. Piskarskas, and P. Foggi, “Powerful sub-100-fs pulses broadly tunable in the visible from a blue-pumped parametric generator and amplifier,” J. Opt. Soc. Am. B 14, 1245-(1997). [CrossRef]
16.	T. Kanai, X. Zhou, T. Liu, A. Kosuge, T. Sekikawa, and S. Watanabe, “Generation of terawatt 10-fs blue pulses by compensation for pulse-front distortion in broadband frequency doubling,” Opt. Lett. 29, 2929–2931 (2004). [CrossRef]
17.	A. Dubietis, G. Valiulis, G. TamoSauskas, R. Danielius, and A. Piskarskas, “Nonlinear pulse compression in the ultraviolet,” Opt. Commun. 141, 50–55 (1997).
18.	J. Hebling, G. Almasi, I. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large area THz-pulse generation,” Opt. Express 10, 1161–1166 (2002). [PubMed]
19.	A. Stepanov, J. Kuhl, I. Kozma, E. Riedle, G. Almási, and J. Hebling, “Scaling up the energy of THz pulses created by optical rectification,” Opt. Express 13, 5762–5768 (2005). [CrossRef] [PubMed]
20.	C. Radzewicz, M. J. la Grone, and J. S. Krasinski, “Interferometric measurement of femtosecond pulse distortion by lenses,” Opt. Commun. 126, 185–190 (1996). [CrossRef]
21.	G. Pretzler, A. Kasper, and K. J. White, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70, 1–9 (2000). [CrossRef]
22.	K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, “High precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B 74, S259–S263 (2002). [CrossRef]
23.	S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE,” Opt. Express 11, 491–501 (2003). [CrossRef] [PubMed]
24.	C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry,” Appl. Phys. B 74, 209–219 (2002). [CrossRef]
25.	Z. Sacks, G. Mourou, and R. Danielius, “Adjusting pulse front tilt and pulse duration by use of a single-shot autocorrelator,” Opt. Lett. 26, 462–464 (2001). [CrossRef]
26.	M. Raghuramaiah, A. K. Sharma, P. A. Naik, and P. D. Gupta, “Simultaneous measurement of pulse front tilt and pulse duration of a femtosecond laser beam,” Opt. Commun. 223, 163–168 (2003). [CrossRef]
27.	G. Figueira, L. Cardoso, N. Lopes, and J. Wemans, “Mirrorless single-shot tilted-pulse-front Autocorrelator,” J. Opt. Soc. Am. B 22, 2709–2714 (2005). [CrossRef]
28.	O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986). [CrossRef]
29.	O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986). [CrossRef]
30.	K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27, 2034–2036 (2002). [CrossRef]
31.	K. Osvay, A. P. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatari, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10, 213–220 (2004). [CrossRef]
32.	S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal distortions of Gaussian pulses and beams,” Opt. Express 13, 8642–8661 (2005). [CrossRef] [PubMed]
33.	Y. Nabekawa and K. Midorikawa, “High-order pulse front tilt caused by high-order angular dispersion,” Opt. Express 11, 3365–3376 (2003). [CrossRef] [PubMed]

OCIS Codes
(140.7090) Lasers and laser optics : Ultrafast lasers
(320.7100) Ultrafast optics : Ultrafast measurements
(320.7160) Ultrafast optics : Ultrafast technology

ToC Category:
Ultrafast Optics

History
Original Manuscript: August 28, 2006
Revised Manuscript: October 29, 2006
Manuscript Accepted: December 10, 2006
Published: December 22, 2006

Citation
Avnish K. Sharma, Rajesh Kumar Patidar, Manchi Raghuramaiah, Prasad Anant Naik, and Parshotam Dass Gupta, "Measuring pulse-front tilt in ultrashort pulse laser beams without ambiguity of its sign using single-shot tilted pulse-front autocorrelator," Opt. Express 14, 13131-13141 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-26-13131

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References

T. Tajima and G. Mourou, "Zettawatt-exawatt lasers and their applications in ultrastrong-field physics," Phys. Rev. Spl. topics accelerators and beams 5, 031301,1-9 (2002). [CrossRef]
J. H. Sung, J. Y. Park, T. Imran, Y. S. Lee, and C. H. Nam, "Generation of 0.2-TW5.5-fs optical pulses at 1 kHz using a differentially pumped hollow-fiber chirped-mirror compressor," Appl. Phys. B Laser & Optics 82, 5-8 (2006). [CrossRef]
Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga and Y. Izawa, "Prepulse-free petawatt laser for a fast ignitor," IEEE J. Quantum. Electron. 40, 281-293 (2004). [CrossRef]
C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt and B. E. Wyborn, "Vulcan petawatt - an ultra-high-intensity interaction facility," Nucl. Fusion 44, S239-S249 (2004). [CrossRef]
M. Aoyama, K. Yamakawa, Y. Akahane, J. Ma, N. Inoue, H. Ueda and H. Kiriyama," 0.85-PW, 33-fs Ti:sapphire laser," Opt. Lett. 28, 1594-1596 (2003). [CrossRef] [PubMed]
Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, "Femtosecond pulse front tilt caused by angular-dispersion," Opt. Eng. 32, 2501-2504 (1993). [CrossRef]
Z. Bor and B. Racz, "Group-velocity dispersion in prisms and its application to pulse-compression and traveling-wave excitation," Opt. Commun. 54, 165-170 (1985). [CrossRef]
J. Hebling, "Derivation of the pulse front tilt caused by angular dispersion," Opt. Quantum. Electron. 28, 1759-1763 (1996). [CrossRef]
X. Gu, S. Akturk, and R. Trebino, "Spatial chirp in Ultrafast Optics," Opt. Commun. 242, 599-604 (2004). [CrossRef]
S. Akturk, X. Gu, E. Zeek, and R. Trebino, "Pulse-front tilt caused by spatial and temporal chirp," Opt. Express 12, 4399-4410 (2004). [CrossRef] [PubMed]
C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, "Temporal aberrations due to misalignments of a stretcher-compressor system and compensation," IEEE J. Quantum. Electron. 30, 1662-1670 (1994). [CrossRef]
K. Osvay, A. P. Kovacs, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, "Measurement of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser," Opt. Commun. 248, 201-209 (2005). [CrossRef]
J. C. Chanteloup, E. Salmon, C. Sauteret, A. Migus, P. Zeitoun, A. Klisnick, A. Carillon, P. Hubert, D. Ros, P. Nickles, and M. Kalachnikov, "Pulse-front control of 15-TW pulses with a tilted compressor, and application to the subpicosecond traveling-wave pumping of a soft-x-ray laser," J. Opt. Soc. Am. B 17, 151-157 (2000). [CrossRef]
A. Shirakawa, I. Sakane, and T. Kobayashi, "Pulse-front-matched optical parametric amplificationfor sub-10-fs pulse generation tunable in the visible and near infrared," Opt. Lett. 23, 1292-1294 (1998) [CrossRef]
P. Di Trapani, A. Andreoni, C. Solcia, G. P. Banfi, R. Danielius, A. Piskarskas, and P. Foggi, "Powerful sub-100-fs pulses broadly tunable in the visible from a blue-pumped parametric generator and amplifier," J. Opt. Soc. Am. B 14, 1245 (1997). [CrossRef]
T. Kanai, X. Zhou, T. Liu, A. Kosuge, T. Sekikawa, and S. Watanabe, "Generation of terawatt 10-fs blue pulses by compensation for pulse-front distortion in broadband frequency doubling," Opt. Lett. 29, 2929-2931 (2004). [CrossRef]
A. Dubietis, G. Valiulis, G. TamoSauskas, R. Danielius, and A. Piskarskas, "Nonlinear pulse compression in the ultraviolet," Opt. Commun. 141, 50-55 (1997).
J. Hebling, G. Almasi, I. Kozma, and J. Kuhl, "Velocity matching by pulse front tilting for large area THz-pulse generation," Opt. Express 10, 1161-1166 (2002). [PubMed]
A. Stepanov, J. Kuhl, I. Kozma, E. Riedle, G. Almási, and J. Hebling, "Scaling up the energy of THz pulses created by optical rectification," Opt. Express 13, 5762-5768 (2005). [CrossRef] [PubMed]
C. Radzewicz, M. J. la Grone and J. S. Krasinski, "Interferometric measurement of femtosecond pulse distortion by lenses," Opt. Commun. 126, 185-190 (1996). [CrossRef]
G. Pretzler, A. Kasper, and K. J. White, "Angular chirp and tilted light pulses in CPA lasers," Appl. Phys. B 70, 1-9 (2000). [CrossRef]
K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, "High precision measurement of angular dispersion in a CPA laser," Appl. Phys. B 74, S259-S263 (2002). [CrossRef]
S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, "Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE," Opt. Express 11, 491-501 (2003). [CrossRef] [PubMed]
C. Dorrer, E. M. Kosik, and I. A. Walmsley, "Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry," Appl. Phys. B 74, 209-219 (2002). [CrossRef]
Z. Sacks, G. Mourou, and R. Danielius, "Adjusting pulse front tilt and pulse duration by use of a single-shot autocorrelator," Opt. Lett. 26, 462-464 (2001). [CrossRef]
M. Raghuramaiah, A. K. Sharma, P. A. Naik, and P. D. Gupta, "Simultaneous measurement of pulse front tilt and pulse duration of a femtosecond laser beam," Opt. Commun. 223, 163-168 (2003). [CrossRef]
G. Figueira, L. Cardoso, N. Lopes, and J. Wemans, "Mirrorless single-shot tilted-pulse-front Autocorrelator," J. Opt. Soc. Am. B 22, 2709-2714 (2005). [CrossRef]
O. E. Martinez, "Pulse distortions in tilted pulse schemes for ultrashort pulses," Opt. Commun. 59, 229-232 (1986). [CrossRef]
O. E. Martinez, "Grating and prism compressors in the case of finite beam size," J. Opt. Soc. Am. B 3, 929-934 (1986). [CrossRef]
K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, "Angular dispersion of femtosecond pulses in a Gaussian beam," Opt. Lett. 27, 2034-2036 (2002). [CrossRef]
K. Osvay, A. P. Kovacs, Z. Heiner, G. Kurdi, J. Klebniczki, M. Csatari, "Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors," IEEE J. Sel. Top. Quantum Electron. 10, 213-220 (2004). [CrossRef]
S. Akturk, X. Gu, P. Gabolde and R. Trebino, "The general theory of first-order spatio-temporal distortions of Gaussian pulses and beams," Opt. Express 13, 8642-8661 (2005). [CrossRef] [PubMed]
Y. Nabekawa and K. Midorikawa, "High-order pulse front tilt caused by high-order angular dispersion," Opt. Express 11, 3365-3376 (2003). [CrossRef] [PubMed]

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