## Abstract

We introduce a novel, chipscale device capable of single-shot ultrafast recording with picosecond-scale resolution over hundreds of picoseconds of record length. The device consists of two vertically-stacked III-V planar waveguides forming a Mach-Zehnder interferometer, and makes use of a transient, optically-induced phase difference to sample a temporal waveform injected into the waveguides. The pump beam is incident on the chip from above in the form of a diagonally-oriented stripe focused by a cylindrical lens. Due to time-of-flight, this diagonal orientation enables the sampling window to be shifted linearly in time as a function of position across the lateral axis of the waveguides. This time-to-space mapping allows an ordinary camera to record the ultrafast waveform with high fidelity. We investigate the theoretical limits of this technique, present a simulation of device operation, and report a proof-of-concept experiment in GaAs, demonstrating picosecond-scale resolution over 140 ps of record length.

© 2011 OSA

## 1. Introduction to device concept

Single-shot optical recording with picosecond-scale resolution across hundreds of picoseconds currently does not have a strong technology base, yet is required for diagnostics in high-energy-density physics experiments [1]. Conventional recording technologies such as streak cameras and oscilloscopes cannot maintain high dynamic range at picosecond-scale resolutions [2]. Single-shot frequency resolved optical gating can achieve a resolution of less than 10 fs but over record lengths of only a few picoseconds [3]. Picosecond-scale resolution has been achieved over hundreds of picoseconds using temporal imaging and time-to-frequency conversion, as well as with single-shot cross-correlators employing pulse front tilt and signal copying [4–7]. Cross-correlators map the structure of a signal from a temporal axis onto a spatial axis, but scaling up record lengths beyond the present limits remains challenging. Chipscale recording methods based on rapidly deflected optical signals have been demonstrated at the picosecond scale via electro-optic [8], and optically-pumped architectures [9]. Time-to-space mapping techniques such as these are highly desired because they can enable a conventional, high fidelity camera to record the serial samples of the signal together in parallel. Here, we introduce a novel device that also maps time to space, but features a much simpler fabrication process than the techniques mentioned above, and could potentially scale to longer record lengths.

Our technique, depicted in Fig. 1
, is termed Synchronously Coupled Anamorphic Light Pulse Encoded Laterally (SCALPEL). The device samples a “slice” of a signal diagonally across time and space. Here, we define the x axis as the spatial axis (along which the signal is uniform), and the z axis as the temporal axis (along which the waveform is spread out by time-of-flight). Temporal gating is achieved by optically inducing a relative phase shift between two planar waveguides stacked vertically (in the y axis) forming a Mach-Zehnder interferometer (MZI). The waveguides are initially biased with a π-phase shift for destructive interference. When a single, short-pulse pump beam of the correct energy is focused with a cylindrical lens to a line focus orthogonal to the planar waveguides, the induced phase difference between the two MZI arms can be made to sweep through 2π radians. This temporarily opens a short window via constructive interference and then closes it again via destructive interference creating a cos^{2} sampling gate. By diagonally orienting the pump stripe at an angle *θ* to the axis of propagation, the window is linearly shifted in time as a function of lateral position across the waveguide. Much like a tilted-pulse front cross-correlator [6], the serial, temporal waveform is thus encoded spatially along the lateral dimension of the planar waveguide for recording in parallel on a camera.

The splitting and combining couplers in the MZI can be fabricated directly into the waveguide epilayers as implied in Fig. 1. Alternatively, the structure can be built without couplers. In this scheme, the signal is end-fire coupled in with a line focus that overlaps both cores, exciting them with equal intensities. The outputs of both waveguides diffract and interfere, creating a fringe pattern of bright and dark rows. This scheme reduces thethroughput efficiency, but greatly simplifies the fabrication and does not otherwise detract from device performance. The MZI arm outputs can be balanced in intensity by adjusting the vertical offset of the signal beam at the input coupling plane and biasing for destructive interference can be achieved by sampling the spatially recorded signal along a null of the vertical fringe pattern recorded on the camera. The pump beam induces a fluence-dependent nonlinear optical phase shift that vertically displaces the later-arriving interference pattern by one fringe and re-stitches it everywhere except at the sliding sampling gate. The null remains dark except where the sliding sampling gate reveals and resolves the signal waveform.

## 2. Gate response and device resolution limits

The pump beam can be characterized by four parameters: its temporal *1/e ^{2}* half width

*w*, its spatial

_{t}*1/e*half width along the

^{2}*z*direction

*w*, its peak fluence

_{z}*F*, and its orientation angle

_{max}*θ*with respect to the axis of propagation. The pump will introduce a change in the relative effective index,

*∆n*, where we denote

^{eff}_{12}(x,z,t)*∆n*as the change in effective index of the top waveguide and

^{eff}_{1}*∆n*as the change in effective index of the bottom waveguide. The form of

^{eff}_{2}*∆n*will be derived in the next section; for now, we can assume that it is known. Figure 2 shows a top view of the interaction of a pump beam with a test-signal consisting of two impulses. The MZI gating response and the change in relative phase shift,

^{eff}_{12}(x,z,t)*∆φ*, are shown, where

_{12}*∆φ*and

_{1}*∆φ*are the phase shifts induced in the top and bottom waveguides, respectively. The relative phase shift incurred between the waveguides as the signal propagates through the pumped region is given as:

_{2}_{$$\Delta {\phi}_{12}(x)=\frac{2\pi}{\lambda}{\displaystyle {\int}_{{z}_{o}}^{\infty}\Delta {n}^{eff}{}_{12}\left(x,z,\frac{z-{z}_{o}}{{v}_{g}}\right)}\text{dz}$$}(2)where

*z*is the position of the impulse at

_{o}*t = 0*, from which the phase shift accumulates, and

*v*is the group velocity. The retardation time is related to the propagation distance as (

_{g}*z-z*)

_{0}*/v*.

_{g}We assume a negative pump-induced index change. When *∆φ _{12}* = -π, the MZI response reaches its maximum (recall that the arms are originally π-phase shifted). When

*∆φ*= π

_{12}_{$\pm $}π/2, the result is ½ of the peak intensity. Because the mapping is linear, each of the diffraction-limited spots shown is equivalent to the impulse response of the device’s time to space mapping. Here, for simplicity, we treat the pump fluence profile as uniform. That is,

*P*is the area occupied by the pump beam. We will further assume that the pump temporal profile is a delta-impulse,

*w*= 0. That is,

_{t}*∆n*, where

^{eff}_{12}(t) = ∆n^{eff}_{12}u(t)*u(t)*is the Heaviside step function.

Consider the points on Impulse 1 at position A in Fig. 2 and all positions above A. After propagating to the output, there should be no change in the relative phase shift because the pump is not experienced by this part of the impulse. The points at position E and below are designed to experience *∆φ _{12} =* −2π, so that the impulse response is localized between points A and E. This constrains the pump’s spatial beam width along the

*z*axis to be

For a given value of sin*θ*, the spatial *1/e ^{2}* half width of the pump stripe can be determined from

*w*, as depicted in Fig. 2. The points between positions A and E experience

_{z}*∆φ*[0,-2

_{12$\in $}*π*];

*∆φ*decreases monotonically with increasing

_{12}*x*, a property that results in a smooth, single-peaked impulse response. The position where

*∆φ*-

_{12}=*π*is denoted by C. The position B experiences

*∆φ*= -π/2, after propagating through a distance of

_{12}*w*through the pumped region. The position D experiences

_{1}*∆φ*= −3π/2, after propagating through a distance of

_{12}*w*through the pumped region.

_{2}The spatial resolution *δx* of the device is defined to be consistent with the Rayleigh criterion for resolvability, which states that two spots are “just resolvable” if they are separated by their 3dB full width at half maximum (FWHM). Positions B and D correspond to the 3dB-points of the impulse response. Thus, the spatial resolution of the device is given by

The temporal resolution *δt* is linearly mapped from *δx* by the orientation angle *θ* of the pump.

In Fig. 2, Impulse 2 is depicted to be advanced in time relative to Impulse 1 by the device temporal resolution *δt*. As a result, the time-to-space mapping of Impulse 2 places its corresponding diffraction-limited spot one resolvable unit *δx* away from the diffraction-limited spot corresponding to Impulse 1.

Using typical semiconductor parameters, we calculate the theoretical lower bound of the temporal resolution *δt* to be 1.1 ps. In general, a larger spatial resolution *δx* is desirable to mitigate diffraction effects and to reduce imaging requirements. The trade-off here is between *δx* and the record length, which is also a function of *θ*. Note that to first order, the time resolution *δt* is not a function of the pump stripe orientation, *θ*.

## 3. Required pump-induced differential phase shift (nonlinear index change)

It is important to distinguish between a change in the effective index of a waveguide, *∆n ^{eff}_{i}(x,z,t)*, and a change in the local refractive index,

*∆n*. The local refractive index change has a

_{i}(x,y,z,t)*y*dependence. The effective index change can be calculated from the local index change by applying first-order perturbation theory to the Helmholtz equation.

*U*is the unperturbed transverse mode profile,

_{i}(y)*n*is the unperturbed effective index, and

^{eff}_{i}*∆ε*. Here,

_{i}(x,y,z,t) = 2n_{core}∆n_{i}(x,y,z,t) + ∆n_{i}^{2}(x,y,z,t)*n*is the unperturbed waveguide core index. The waveguides are designed so that the pump photon energy is larger than the bandgap of the core but smaller than the bandgap of the cladding. Thus,

_{core}*∆n*is negligible in the waveguide cladding. The photon energy of the signal is smaller than the bandgap of the waveguide cladding and core materials.

_{i}(y,t)The dynamics of the material response due to the pump occurs on the femtosecond to nanosecond scale, and can be modeled to have the form [10]

where the operation_{$\otimes $}is the convolution,

*I*is the instantaneous intensity, and

*F*is the pump fluence. Or, at a certain position of interest

_{o}*(x,z)*

_{$\in $}*P*,

The intensity of a Gaussian pulse can be described as

The term *h(y,t)* is the impulse response of the refractive index change. The term *n _{g}* is the group index of the pump pulse propagating in the planar waveguide. The temporal 1/e

^{2}half width,

*w*, is generally a few hundred femtoseconds. The time dependence of

_{t}*h(y,t)*is a complicated expression that depends on the contributions from spectral hole burning (SHB), carrier heating, the increase in carrier density, and mechanisms much faster than the pump width, such as two photon absorption (TPA) and the optical Stark effect (OSE) [10].

We will ignore the effects of SHB, TPA, and OSE since they disappear within a fraction of a picosecond, much shorter than the timescales of interest as it affects the signal (1-200 ps). Carrier heating, despite having a time constant on the order of a picosecond, can be ignored as well, because a significant portion of carrier heating comes from free carrier absorption, which is small in intrinsic III-V semiconductors such as GaAs [10-11]. Only the carrier density term has a lifetime of nanoseconds. Thus, we make the approximation

where ∆*n*is the value of the index change associated with the change in carrier concentration ∆

_{ρ}(y)*ρ(y)*. Substituting in the simplified

*h(y,t)*, Eq. (10) can be written

By working only with ∆*n _{ρ}(y)* and ignoring the faster dynamics, our closed-form description of the refractive index change is conservative, in the sense that it underestimates the magnitude of

*∆n*. Carrier heating and SHB generally decrease the refractive index for photon energies below the bandgap. Two-photon absorption and OSE provide negative and positive contributions to the refractive index change, respectively. In actuality, the transient is not monotonic as described by Eq. (13); rather it overshoots and then quickly converges [10]. However, simulations suggest that this transient has only a small impact on the resolution of the device.

_{i}(y,t)To model the index change due to the increase in carrier concentration, we first find the change in carrier concentration ∆*ρ(y)*, which is approximately the carrier concentration induced by the pump *ρ(y)*, since intrinsic GaAs has a relatively small number of free carriers in steady state.

Because the pump fluence needed to introduce the requisite index change is quite large, the absorption of the pump in the waveguide core cannot simply be described by Beer’s Law. The intensity of the pump follows the pair of differential equations

*β*is the TPA coefficient and α is the single photon absorption (SPA) coefficient. Notice that we cannot ignore TPA and SHB here since the relevant timescales for pump absorption is much shorter than that of signal propagation. The pump is absorbed in waveguides that are 0.6 μm thick, while the signal experiences the index change over a distance that is 50 to 100 times longer. Over these timescales, spectral hole burning bleaches the absorption at the pump photon energy. To first order, the SPA coefficient

*α*at the pump wavelength is proportional to the density of available states,

*ρ*-

_{max}*ρ*, where

*ρ*is the population inversion carrier density threshold and

_{max}*ρ*is the carrier density due to SPAwhere α

_{o}is the non-saturated absorption coefficient. Neglecting the possible reshaping of

*I(y,t)*due to nonlinearities, Eq. (15) and 16 can be solved simultaneously for the free carrier distribution [12]

The first term in the summation is due to SPA, while the second term is due to TPA. The constant *r* is a pulse-shape-dependent numerical constant (*r* = 1 for a rectangular pulse, *r* = 0.67 for a Gaussian pulse) [12]. *F _{s}* is the saturation fluence

*F(y)* is the pump fluence that passes through a thin layer of semiconductor at depth *y*

The carrier distribution is plotted on a log scale for different pump fluences in Fig. 3(a) . Absorption saturation can be observed for increased pump fluences, as indicated by the flattening of the carrier density profile.

The value of ∆*n _{ρ}* can be attributed to three main carrier density dependent effects: bandfilling, bandgap shrinkage, and the plasma loading effect, also known as free carrier absorption. Bandfilling describes the decrease in absorption due to the population of conduction states and depopulation of valence states, and through the Kramers-Kronig integral, results in a decrease in refractive index for photon energies slightly below the bandgap [13]. Bandgap shrinkage shifts the absorption edge to lower energies, thus increasing absorption below the bandgap, resulting in a positive change in refractive index proportional to

*ρ*[14]. The change in refractive index due to the plasma loading effect, or free carrier absorption, is due to intraband effects that can be described by the Drude oscillator model to be proportional to

^{1/3}*ρ*[15]. For wavelengths slightly above the bandgap wavelength, ∆

*n*is dominated by the effects of bandfilling. We numerically calculate ∆

_{ρ}*n*as a function of wavelength and carrier density and use Eq. (8) to determine

_{ρ}(y)*∆n*and

^{eff}_{1}*∆n*, which are shown in Fig. 3(b) along with

^{eff}_{2}*∆n*as a function of the pump fluence [16]. A maximum value of

^{eff}_{12}*∆n*0.005 is achieved for a pump fluence of 60 μJ/cm

^{eff}_{12}=^{2}, limited ultimately by absorption saturation. The optimal temporal resolution is correspondingly calculated to be 1.1 ps. Using Eq. (13), the transient can be obtained by taking

*I(t)*to be of the form described in Eq. (11).

*t = 0*is when the pump has maximum intensity. We can assume that the impact of the pump pulse width on the temporal resolution is negligible when

*w*. Temporal resolution is not very sensitive to the pump pulse width: for example, a pump pulse width of 1 ps will degrade the resolution by 0.3% compared to a pulse width of zero.

_{t}« w_{1}/v_{g}Up to this point, we have been working within the regime of hundreds of picoseconds, where *t* is less than the free carrier lifetime. For the purposes of this analysis, free carrier diffusion will be neglected. However, for completeness, if the carrier distribution is uniform within the waveguides, due to diffusion, Eq. (8) can be simplified to

*∆n*is the index change associated with the uniform carrier density. At room temperature, for undoped GaAs, the diffusion coefficient is approximately

^{unif}_{i}*D*= 200 cm

^{2}/s and the diffusion length is approximately 4 μm in 200 ps [17]. In Fig. 3(b), the dotted blue curve represents

*∆n*for a uniform carrier distribution. As carriers diffuse within the waveguide core, dotted red curve of Fig. 3(b) will evolve into the dotted blue curve.

^{eff}_{12}## 4. Record length limitations: illumination length and diffraction

Recall from Fig. 2 that the relative phase shift ∆φ_{12} due to the pump stripe causes the signal waveform to be gated out, effectively mapping time to space. In the previous section, the form of *∆n ^{eff}_{12}* was derived. With

*∆n*, the spatial resolution and temporal resolution can be found via Eqs. (6) and (7). Another parameter of interest is the record length of the device, measured in time. Neglecting diffraction, the record length is limited by the orientation of the pump stripe and the geometry of the chip. Figure 4 depicts the spatial resolution (green solid curve) and the record length (blue dotted curve) as a function of θ, for the maximum value of

^{eff}_{12}*∆n*from Fig. 3(b).

^{eff}_{12}Diffraction and refraction of the signal beams interacting with the index stripe can introduce artifacts that limit device performance. For the beam sizes of interest, the Raleigh range of collimated input beams is much longer than the device length. The depth of field associated with the gated beamlets also does not compromise resolution. However, a wavefront tilt artifact introduced inside the stripe can impact the resolution at the edges of the record as illustrated in Fig. 5(a) . This will limit the record length of the device because samples later in time must propagate further after experiencing the pump stripe on route to the camera.

We simulate the effect of the local wavefront tilt with a split-step beam propagation method (BPM). The angle is chosen to be 12°, corresponding to the edge of the roll-off seen in the record length of Fig. 4. Figure 5(b) shows from a top view its effects on the propagation of a single impulse as it interacts with the pump stripe (outlined in black). The effect here looks dramatic, but because both waveguides experience the same tilt, its effect is diminished at the recording plane. Figure 5(c) is a simulation of a series of impulses separated in time by the temporal resolution. Towards the end of the record, the samples are distorted and decrease in intensity. We define the end of the record to be where the impulse response peak has dropped to ½ of the peak at the beginning of the record. The record length calculated with diffraction is plotted as squares in Fig. 4(a). We expect diffraction to limit the record length for small *δx*, while for large *δx* the record length is limited by the device dimensions.

## 5. Experimental

The SCALPEL waveguides were grown by metal-organic chemical vapor deposition (MOCVD) on a GaAs substrate and consisted of two GaAs planar waveguide cores with Al_{0.24}Ga_{0.76}As cladding layers, the details of which are depicted in Fig. 6
. The structure was built without splitting and combining couplers, which greatly simplifies the fabrication process: after cleaving the wafers to the operating size of 5 cm (*z*) by 1 cm (*x*) with optical quality input/output facets, no further processing was necessary. Eliminating the integrated couplers decreases the throughput efficiency but does not otherwise detract from device performance. The signal is end-fire coupled with a line focus that overlaps both cores, exciting them with equal intensities. The outputs of both waveguides diffract and interfere, creating a fringe pattern of bright and dark rows. The signal was coupled into and out of the two waveguides with the help of fast-axis collimating microlenses. The interference pattern at the waveguide output was captured on a CCD camera. To obtain the signal trace, a lineout was taken in *x* along the first (zero-order) null of the interferogram integrated on the camera. Note that operating on this zero-order null yields an octave-spanning spectral bandwidth over which fringe smearing is negligible—much larger than the 8 nm bandwidth of the signal.

The pump was a 380 fs (FWHM) pulse at 804 nm generated by a Ti:Sapphire-based ultrafast oscillator and 1 kHz regenerative amplifier. The pump has a photon energy above the bandgap of the cores but below that of the cladding layers. The pump beam was focused into an approximately 30-60 μm (FWHM) diagonal stripe by a cylindrical lens oriented at a variable angle *θ* with respect to the z axis. A test signal was generated by diverting part of the pump energy to drive an optical parametric amplifier that produced an idler beam at 1.9 μm, which was subsequently frequency-doubled to 950 nm, which is just below the bandgap of the waveguide cores so that it experiences a strong carrier-induced index change. The lateral width of the signal beam was 10 mm. A silicon CCD camera captured the output fringe pattern. A test signal at 950 nm was generated by a Michelson interferometer, yielding two 170 fs (FWHM) impulses separated by 124 ps. The experimental setup is illustrated in Fig 7(a)
.

A single-shot recording of the test signal is shown in Fig. 7(b). In this case, the angle of the pump stripe *θ* is set to 12°, corresponding to the edge of the roll-off in the record length when no diffraction is assumed, from Fig. 4. This is achieved by formatting the pump beam into an approximately 60 μm (full width *1/e ^{2}*) diagonal Gaussian stripe providing a peak fluence of 275 μJ/cm

^{2}. For a Gaussian pump beam, the fluence is not uniform in

*z*, which means

*∆n*is not constant with

^{eff}_{12}*z*, nor is it a nice single-peaked function of

*z*, in general. The Gaussian beam does not degrade the resolution with respect to a square uniform beam if we operate at the regime near the optimal temporal resolution.

For the SCALPEL technique to work properly, both the carrier recombination time (nanoseconds) and the integration time of the camera (100s of microseconds) should be longer than the time record of interest (100s of picoseconds) but shorter than the signal repetition rate (here a millisecond for the kHz source). The theoretical resolution for these pump parameters is approximately 3 ps. The resolution was measured to be 7.5 ps, most likely limited by group velocity dispersion (GVD), as the bandwidth of the signal pulse is 8 nm (2.6 THz). These effects can be mitigated by spectrally filtering the input signal and introducing an absorbing layer between the waveguide cores to increase the relative index change experienced by the MZI arms. The record length was limited to 140 ps by the defect-free area of the cleaved facets. Over shorter record lengths, measurements with 2 ps resolution were achieved.

## 6. Conclusion

SCALPEL device models indicate that a resolution of ~1 ps with ~200 ps of record length is achievable. The resolution is limited by the difference in index change between the two arms of the MZI, which is limited by absorption saturation in the current design. Introducing an absorbing layer between the waveguide cores will reduce the index change experienced by the lower MZI arm and further improve the resolution. Alternatively, introducing a multilayer Bragg reflector between the waveguide cores will reduce the index change experienced by the lower arm, while recycling the pump energy through the upper arm. This could improve the resolution to 0.7 ps. More sophisticated methods of analyzing the interference fringe pattern may help to increase the signal to noise ratio of the processed trace.

We have introduced a device capable of single-shot optical recording with near-picosecond resolution and 100s of picoseconds of record length. The device maps the time content of the signal waveform onto the lateral space axis, enabling the serial, temporal signal to be recorded in parallel with a conventional, high fidelity camera. The technique is a hybrid between single-shot cross-correlation techniques developed by the ultrafast community and interferometric switches developed by the integrated optics community. It retains their advantages but eliminates many of their disadvantages. The record lengths of single-shot cross-correlators are limited by the crystal aperture size and the need to impart pulse-front tilt requiring large diffraction gratings which unavoidably impart undesirable angular dispersion [18]. In contrast, SCALPEL’s record length can be extended by simply rotating the focused pump stripe and scaling to larger wafers (up to the diffraction limit). Interferometric switches are fabrication intensive and plagued by inconsistent gate outputs and extinction, as well as high scattering losses. In contrast, fabrication of a SCALPEL device is lithography- and etch-free, naturally avoiding many of these unwanted artifacts.

## Acknowledgments

We acknowledge the useful discussions and support from Mark Lowry. This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

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