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Numerical simulations of ultrasimple ultrashortlaser-pulse measurement
Optics Express, Vol. 15, Issue 8, pp. 4585-4596 (2007)
http://dx.doi.org/10.1364/OE.15.004585
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– Abstract
1. Ultrashort-laser-pulse device complexity and its reduction: GRENOUILLE
2. Numerical Simulation of GRENOUILLE
3. Results
4. Conclusions
– References and links
– Abstract
1. Ultrashort-laser-pulse device complexity and its reduction: GRENOUILLE
2. Numerical Simulation of GRENOUILLE
3. Results
4. Conclusions
– References and links
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Abstract
We numerically simulate the performance of the ultrasimple frequency-resolved-optical-gating (FROG) technique, GRENOUILLE, for measuring ultrashort laser pulses. While simple in practice, GRENOUILLE has many theoretical subtleties because it involves the second-harmonic generation of relatively tightly focused and broadband pulses. In addition, these processes occur in a thick crystal, in which the phase-matching bandwidth is deliberately made narrow compared to the pulse bandwidth. In these simulations, we include all sum-frequency-generation processes, both collinear and noncollinear. We also include dispersion using the Sellmeier equation for the crystal BBO. Working in the frequency domain, we compute the GRENOUILLE trace for practical—and impractical—examples and show that accurate measurements are easily obtained for properly designed devices.
© 2007 Optical Society of America
1. Ultrashort-laser-pulse device complexity and its reduction: GRENOUILLE
Measuring ultrashort laser pulses has traditionally been a difficult task. Virtually
all available techniques are based on autocorrelation, which requires splitting the
pulse into two replicas, recombining them in space and time in a nonlinear-optical
medium, and measuring the nonlinear-optical signal pulse while varying the delay
between the two replicas. Methods that yield more than the mere autocorrelation also
require additional optics, such as a spectrometer, and some methods also involve
devices as complex as interferometers and pulse shapers or stretchers, as well.
Complex devices are inherently difficult to work with and are usually easily
misaligned. As a result, complex devices often introduce the very distortions they
are designed to measure. While complex computer programs can also be required, this
does not add complexity to the device operation, as, in the case of
frequency-resolved-optical-gating (FROG), such programs are available commercially
and are now also very reliable and fast. And computer programs do not misalign or
change with time. Thus, experimental simplicity is the high
priority of ultrashort-pulse measurement.
Recently, an extremely simple pulse-measurement device was introduced and is now in
wide use. A highly simplified version of the FROG [1] family of devices, this method (called GRENOUILLE [2]; see Fig.1) operates by using a simple, large-apex-angle prism (a
“Fresnel biprism”) to split the beam into two replicas and to
automatically cross and align them in space and time in the crystal. It uses a
relatively large beam-crossing angle and a line focus, so the pulse
replicas’ relative delay is mapped onto the crystal transverse position
(see Fig. 2(a)) yielding single-shot operation.
GRENOUILLE’s second innovation is the use of a thick
second-harmonic-generation (SHG) crystal[2,3], which, due to its thickness, phase-matches only a
small—and different—fraction of the pulse bandwidth for each
output angle, allowing the crystal to operate, not only as an autocorrelating
element, but also as the dispersive element of a spectrometer (see Fig. 2(b)). The phase-matching bandwidth of the thick crystal
is then GRENOUILLE’s spectral resolution—not
its spectral range, as in other pulse-measurement devices. The
spectral range is determined instead by the angular divergence of the beam.
P. O’Shea, M. Ki1mmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. 26, 932–934(2001). [CrossRef]
P. O’Shea, M. Ki1mmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. 26, 932–934(2001). [CrossRef]
C. Radzewicz, P. Wasylczyk, and J. S. Krasinski, “A poor man’s FROG,” Opt. Comm. 186, 329–333 (2000) [CrossRef]
These two innovations yield a very simple, compact FROG device composed of only four
easily aligned, linearly arranged, optical elements and that requires almost no
alignment and never misaligns. In addition, without modification, GRENOUILLE also
measures the spatio-temporal distortions, spatial chirp and pulse-front tilt: the
otherwise symmetrical trace develops shear in the presence of spatial chirp and
displacement along the delay axis in the presence of pulse-front tilt[4, 5].
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]
S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring spatial chirp in ultrashort pulses using single-shot Frequency-Resolved Optical Gating,” Opt. Express 11, 68–78 (2003). [CrossRef] [PubMed]
Fig. 1. (a). FROG (top) and its simpler cousin, GRENOUILLE (bottom). GRENOUILLE
replaces the beam splitter and recombining apparatus with a Fresnel biprism.
And it also replaces the thin crystal and spectrometer with a thick
crystal.
Fig. 2. (a). The Fresnel biprism and its use for splitting and crossing two replicas
of the pulse to be measured. It maps delay onto transverse position of the
crystal.
Fig. 2. (b). Rough polar plots of the output SHG intensity of a given color vs. angle
for a tightly focused broadband input pulse and SHG crystals of various
thicknesses. The thick crystal autocorrelates the tightly focused input
pulse and simultaneously angularly disperses the resulting second-harmonic
pulse.
As with all innovations, the Fresnel biprism and the thick crystal involve some subtlety[6]. For example, the Fresnel biprism should not be used for
extremely short or long pulses. For short pulses, it can introduce too much group
delay dispersion (GDD) and so must be replaced with a “Fresnel
bimirror.” For long pulses, a larger crossing angle is required to
achieve the large range of delays, and the biprism apex angle becomes too small
(i.e., far less than 180°), introducing too much angular dispersion in to
the beams, and the beam may see very different GDD for different transverse
positions. Its replacement with a Fresnel bimirror also has (geometrical)
beam-crossing issues for long pulses. Nevertheless, the Fresnel biprism’s
physics is relatively simple, and it works well for a wide range of pulse lengths
(∼ 20 fs to ∼ 1 ps), so we will not consider its effects here.
P. O’Shea, S. Akturk, M. Kimmel, and R. Trebino, “Practical issues in ultra-short-pulse measurements with ‘GRENOUILLE’,” Appl. Phys. B 79, 683–691 (2004). [CrossRef]
The thick crystal must also be used with care. Use of a thicker crystal yields better
spectral resolution due to a smaller phase-matching bandwidth, but it also
introduces more GDD into the pulse to be measured. Use of a thinner crystal does the
opposite. Fortunately, the appropriate crystal thickness scales with the pulse
length: a thinner crystal has less GDD and less spectral resolution, both
appropriate for a shorter pulse. Nevertheless, the choice of the crystal thickness
in GRENOUILLE is a careful compromise between opposing distortions. But just how
touchy is this compromise? Specifically, how large a range of pulse lengths can be
measured with a given crystal thickness? Ideally, the range should be about an order
of magnitude or more for a single device.
In order to quantitatively answer this question, we must realize that the simple
picture of Fig. 2(b) oversimplifies the potentially complex nonlinear
optics somewhat. Not only is SHG occurring in the thick crystal, but a wide range of
sum-frequency-generation (SFG) processes—both collinear and
noncollinear—are also. So we must include these additional processes and
check that this simple picture based only on SHG ideas accurately reflects the
device’s reality.
So in this paper we will numerically simulate GRENOUILLE measurements of ultrashort
laser pulses, taking into account the above effects involving the thick crystal and
the various SFG processes.
2. Numerical Simulation of GRENOUILLE
To simulate the second-order nonlinear-optical processes in GRENOUILLE, we work in
(k⃗,ω) space and assume
the input beam is aligned along the z-axis, which is perpendicular to the crystal
face. Because the input pulse is broadband, it involves a wide range of input
frequencies (call a given pair ω
1 and
ω
2) generating sum frequencies
ω
3=ω
1
+ ω
2 in a wide range of
directions. Also, because the beams involved are tightly focused, each of these
processes can also occur through off-axis phase-matching processes,
k⃗1 +
k⃗2 =
k⃗3, where
k⃗3 is the phase-matched sum frequency
k-vector in a given direction.
Let y be the transverse direction in which the crystal’s
phase-matching wavelength varies. To simulate the physics of GRENOUILLE, we
calculate the spectrum at the crystal exit face for each time delay and output angle
in the yz-plane. For the x-dimension (the
direction in which the delay between the two beams varies), we simply include a
delay between the two pulse replicas. Under the non-depleted-pump assumption, we
integrate the wave equation:
where E
3 is the sum-frequency field and
E
1,2 correspond to the fundamental input fields.
deff
is the effective nonlinearity and
ñ is the effective refractive index. The field
envelopes are constructed on a grid of (ky
,
ω). The above-mentioned constraints
k
3y
=
k
1y
+
k
2y
and
ω
3 =
ω
1 +
ω
2 are strictly enforced. The
polarization at the generated sum frequency is calculated for each
(ky
, ω) taking into
account the contribution from multiple
E
1(k
1y
,
ω) and
E
2(k
2y
,
ω) pairs that satisfy those conditions. The phase
mismatch Δkz
is complicated and becomes a
function of both frequency and off-axis angle with respect to z.
For a Type I phase-matching process, considering a specific point
q
(k
3y
,
ω
3) on the grid (see Fig. 3), if we assume one of the suitable pairs that
contributes is
P1(k
1y
,
ω
1) and
P2(k
2y
,
ω
2), the phase mismatch along
z-axis Δkqz
for the grid
point q
(k
3y
,
ω
3) is computed by:
The angles θ
1,
θ
2 and
θ
3 can be easily calculated since they
are fixed for each corresponding (ky
,
ω) grid point. For the refractive indices,
ne
and no
, we used
the full Sellmeier equation, so the crystal dispersion is included to all orders,
rather than by an expansion in a power series. Then the radiation field contributed
by P1(k
1y
,
ω
1) and
P2(k
2y
,
ω
2) is simply ∝
i
E
p1
E
p2
(exp(-iΔkqzL)-1)/Δkqz
,
where L is the length of the crystal.
Fig. 3. Diagram for the phase-mismatch calculation. The k-vector of grid
q
(k
3y
,
ω
3) is tilted from the
z-axis by θ
3.
θ
1 and
θ
2 are the tilt angles of the
k-vectors of the electric field pair
P1(k
1y
,
ω
1) and
P2(k
2y
,
ω
2).
For our simulations, we use various temporal fields to test the device, but, in all
cases, we assume a Gaussian-shaped spatial input-field profile. In addition, all the
input beams are assumed to have their waists at the center of the crystal. For each
delay between the two replicas, we integrate Eq. (1) to compute the SHG/SFG field. Because GRENOUILLE involves
interpreting a given crystal output angle as the SHG frequency, we must compute the
SHG/SFG intensity vs. crystal output angle, integrating over all frequencies. And
because the focusing can be tight, we cannot simply interpret the transverse
k-vector component, ky
, as the output angle. In other
words, the off-axis k-vector component yields an output angle that also depends on
the k-vector magnitude (sin θ =
ky
/[ne
(ω,
θ)ω/c]).
To calibrate our simulated GRENOUILLE traces, we simulate a double pulse trace
GRENOUILLE (FROG) trace, which has both well-known temporal and spectral structure
that depends only on the pulse separation, as is done to calibrate GRENOUILLEs in practice[6]. The FROG retrieval algorithm is then applied to the
computed GRENOUILLE trace, and the retrieved pulses are compared with the precise
known input pulses.
P. O’Shea, S. Akturk, M. Kimmel, and R. Trebino, “Practical issues in ultra-short-pulse measurements with ‘GRENOUILLE’,” Appl. Phys. B 79, 683–691 (2004). [CrossRef]
We compare our simulated traces with ideal FROG traces, computed using the well-known
formula:
We also run the standard SHG FROG algorithm for our simulated GRENOUILLE traces and
determine the accuracy with which GRENOUILLE determines both the SHG FROG trace and
also, more importantly, the actual pulse. Our grid size for all traces is 128 x
128.
3. Results
3.1 Does GRENOUILLE yield the correct trace?
To determine whether GRENOUILLE correctly measures pulses in general, we
performed simulations of several test pulses. We present a typical result here
in which we simulated the GRENOUILLE trace of a 60-fs, flat-phase Gaussian,
800-nm input pulse (whose ideal FROG trace is shown in Fig. 4(a)). We use a 3.5 mm BBO SHG crystal, which is
commonly used to measure pulses from 50 to about 500 fs in length. The input
beam was focused to 10 μm in the center of the crystal. The delay
increment used was 7.5591 fs, and the wavelength spacing was 0.5138 nm. The
resulting GRENOUILLE trace is shown in Fig. 4(b), and the corresponding retrieved trace is shown
in Fig. 4(c). The retrieved temporal and spectral
intensities and phases show excellent agreement with the actual pulse temporal
and spectral intensities and phases in Fig. 4(d) and (e). We find excellent agreement among the ideal,
simulated, and retrieved traces. The root mean square (rms) error between the
simulated GENOUILLE trace and the FROG trace was 0.007051. The rms error between
the retrieved GRENOUILLE trace and the FROG trace was 0.006105. The rms error
between the simulated and retrieved GRENOUILLE traces was 0.003534. In other
words, the simulated and retrieved traces are quite accurate. Also, note that
the rms error between the retrieved trace and the ideal FROG trace is less than
that between the simulated trace and the ideal FROG trace. This is because the
FROG algorithm is able to correct for slight discrepancies in the simulated
(measured) trace due to the redundancy in the time-frequency-domain trace. We
find that this occurs in all of our simulations and is a convenient feature of
FROG (an effect we observe experimentally as well).
Fig. 4. (a) Ideal FROG trace for the 60fs flat phase pulse. (b) Simulated
GRENOUILLE trace of a. (c) Retrieved GRENOUILLE trace. (d,e) The black
lines show the retrieved temporal and spectral intensities and phases of
the pulse. The red lines show the intensities and phases of the actual
input pulse.
We also tested GRENOUILLE’s ability to measure complex pulses, and we
present one such example here. Figure 5 shows the ideal FROG trace, the simulated
GRENOUILLE trace, and the retrieved trace for two overlapping chirped 50-fs
pulses with identical parabolic phases (a “double chirped”
pulse) using a 3.5 mm BBO crystal and 10 μm focal spot.
The delay spacing was 9.4488 fs and wavelength spacing was 0.2509 nm. The rms
error between the simulated GENOUILLE trace and the FROG trace was 0.006429. The
rms error between the retrieved GRENOUILLE trace and the FROG trace was
0.006191. The rms error between the simulated and retrieved GRENOUILLE trace was
0.003383. We find that the simulated trace yields an intensity and spectrum that
do not perfectly match the actual pulse, likely due to the finite beam
divergence at the crystal, which yields a slight cropping of the spectrum, but
they are not far off.
Fig. 5. (a) Ideal FROG trace of a double chirped 50 fs pulse. (b) Simulated
GRENOUILLE trace. (c) Retrieved GRENOUILLE trace. (d,e) The black curves
show the retrieved temporal and spectral intensities and phases of the
pulse. The red curves show the intensities and phases of the actual
input pulse.
3.2 Focusing issues
In GRENOUILLE, the thick crystal functions as the spectrometer dispersive
element. Phase-matching maps SHG wavelength to output angle. This means that, in
order to measure pulses with large bandwidths, the beam must have a large
angular divergence, so a tight focus is required. A beam with too large a spot
size and hence too little divergence will cause frequencies at the edges of the
spectrum to be too weak in the resulting GRENOUILLE trace.
In the above simulation for the 60-fs pulse, if we were to use a larger focal
spot with a correspondingly smaller angular divergence instead, an (erroneous)
narrower spectrum would be obtained. And we see precisely this in our
simulations, as shown in the movie in Fig. 6.
Fig. 6. (2.45 MB) Movie of simulated GRENOUILLE traces for a 60-fs flat-phase
pulse focused at the center of a 3.5 mm BBO with different focal-length
lenses. In the movie, the focal spot size evolves from 10 to 100
μm. The weaker foci yield traces that are spectrally too
narrow. [Media 1]
3.3 Crystal thickness
Using a thicker crystal decreases the phase-matching bandwidth and so increases
the GRENOUILLE spectral resolution. But the crystal should also not be too
thick, or the pulse will spread in time due to the crystal GDD, and the pulse
temporal structure will be lost. We simulate this effect by varying the crystal
thickness and watching the simulated GRENOUILLE trace vary (See Fig. 7.). We use the same pulse as in Fig. 5.
Fig. 7. (2.45 MB) Movie of simulated GRENOUILLE trace for a 50-fs double chirped
pulse focused down to the center of a BBO crystal with a
10-μm focal spot. In the movie, the thickness of the BBO
crystal changes from 0.5 mm to 9.5 mm. [Media 2]
Using a 0.5 mm BBO, the spectral fringes are almost completely lost. The temporal
structure of the pulse becomes difficult to recognize for crystals longer than
7.5 mm. Some of the spectral side lobes in the FROG trace (See Fig. 5(a)) become quite fuzzy in the simulated GRENOUILLE
trace, as expected from a low-resolution spectrometer, ∼ 2nm for this
specific GRENOUILLE design.
3.3 Measurement of long and short pulses
The GRENOUILLE with the above mentioned configuration usually measures pulses
∼ 50 to ∼ 500 fs long. However, we find that measurements
of pulses longer than 500 fs are also possible with very good accuracy. Figure 8 shows a case of a double chirped long pulse with
structure in both delay and frequency and with pulse length of ∼550
fs. A 3.5-mm BBO crystal was used with a 10-μm focal spot. The delay
increment was 32.126 fs and the wavelength increment was 0.1589 nm.
Interestingly, GRENOUILLE measures this pulse quite well. The rms error between
the simulated GENOUILLE trace and the FROG trace was 0.021896, quite good for a
pulse this complex. The rms error between the retrieved GRENOUILLE trace and the
FROG trace was 0.015758. The rms error between the simulated and retrieved
GRENOUILLE trace was 0.013183. Again, it appears that the well-known fundamental
redundancy in the time-frequency-domain trace and the robust FROG
phase-retrieval algorithm are able to compensate for the insufficient spectral
resolution in the GRENOUILLE trace.
Fig. 8. (a) Ideal FROG trace of a double chirped long pulse. (b) Simulated
GRENOUILLE trace of the same pulse. (c) Retrieved GRENOUILLE trace.
(d,e) The black curves show the retrieved temporal and spectral
intensities and phases of the pulse. The red curves show the intensities
and phases of the actual input pulse.
The shortest pulses ever measured by GRENOUILLE are ∼20 fs long [7]. The thickness of the BBO crystal in such measurements
was 1.5 mm. A tighter focus ensured the larger divergence angle required to
cover the larger spectrum. The following simulation demonstrates
GRENOUILLE’s ability to measure pulses as short as 20 fs, which also
have with fine structure (See Fig. 9.). We used a 5-μm focal-spot diameter.
The delay increment used was 3.7795 fs and the wavelength spacing used was
1.4509 nm. The rms error between the simulated GENOUILLE trace and the FROG
trace was 0.010379. The rms error between the retrieved GRENOUILLE trace and the
FROG trace was 0.006799. The rms error between the simulated and retrieved
GRENOUILLE trace was 0.008367. The intensities and phases versus time and
frequency retrieve quite well. Pulse distortion due to material dispersion is
negligible. The minor discrepancy is due to slightly insufficient resolution of
the thick crystal ‘spectrometer.’
S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Extremely simple device for measuring 20 fs pulses,” Opt. Lett. 29, 1025–1027 (2004). [CrossRef] [PubMed]
Fig. 9. (a) Ideal FROG trace of a slightly chirped 20 fs double pulse. (b)
Simulated GRENOUILLE trace of this pulse. (c) Retrieved GRENOUILLE
trace. (d,e) The black curves show the retrieved temporal and spectral
intensities and phases of the pulse. The red curves show the intensities
and phases of the actual input pulse.
This specific GRENOUILLE design has a resolution of ∼ 4 nm at 800 nm,
somewhat less than that required to resolve this relatively long pulse. Again,
the FROG retrieval algorithm improves the trace, this time significantly (a
factor of two improvement in the rms error), retrieving the spectral side lobes
reasonably well.
For even shorter pulses, temporal broadening due to dispersion will be a problem
(For BBO, 800nm type I phase matching, GVM(group velocity mismatch) = 1.92
× 103 fs/cm. The GDD(group delay dispersion)= 195.9
fs2/mm at 400nm). However, a thinner crystal could be used in
this case.
4. Conclusions
We have numerically simulated the performance of GRENOUILLE, which involves
considering the complex sum-frequency generation of tightly focused, broadband input
beams in a thick SHG crystal. We take into account dispersion using the full
Sellmeier equation. We have shown that using an appropriate crystal thickness and
beam focus assures the accuracy of a GRENOUILLE measurement.
Specifically, our simulations show that GRENOUILLE is able to accurately measure
pulses over at least an order of magnitude range of pulse lengths, spectral widths,
and temporal and spectral structure. Despite its experimental simplicity, it is even
capable of measuring complex pulses with time-bandwidth products approaching
∼10. Only more complex, complete versions of FROG (and its cousin XFROG)
can do better. Such performance, which matches that of GRENOUILLEs observed
experimentally, is more than adequate for monitoring the output of
today’s ultrafast lasers and even measuring some shaped pulses.
References and links
R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic Publishers, Boston, 2002). | |
P. O’Shea, M. Ki1mmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. 26, 932–934(2001). [CrossRef] | |
C. Radzewicz, P. Wasylczyk, and J. S. Krasinski, “A poor man’s FROG,” Opt. Comm. 186, 329–333 (2000) [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] | |
S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring spatial chirp in ultrashort pulses using single-shot Frequency-Resolved Optical Gating,” Opt. Express 11, 68–78 (2003). [CrossRef] [PubMed] | |
P. O’Shea, S. Akturk, M. Kimmel, and R. Trebino, “Practical issues in ultra-short-pulse measurements with ‘GRENOUILLE’,” Appl. Phys. B 79, 683–691 (2004). [CrossRef] | |
S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Extremely simple device for measuring 20 fs pulses,” Opt. Lett. 29, 1025–1027 (2004). [CrossRef] [PubMed] |
OCIS Codes
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(320.7100) Ultrafast optics : Ultrafast measurements
ToC Category:
Ultrafast Optics
History
Original Manuscript: January 22, 2007
Revised Manuscript: March 26, 2007
Manuscript Accepted: March 27, 2007
Published: April 3, 2007
Citation
Xuan Liu, Rick Trebino, and Arlee V. Smith, "Numerical simulations of ultrasimple ultrashortlaser-pulse measurement," Opt. Express 15, 4585-4596 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-4585
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References
- R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic Publishers, Boston, 2002).
- P. O'Shea, M. Kimmel, X. Gu, and R. Trebino, "Highly simplified device for ultrashort-pulse measurement," Opt. Lett. 26, 932-934(2001). [CrossRef]
- C. Radzewicz, P. Wasylczyk and J. S. Krasinski, "A poor man's FROG," Opt. Commun. 186,329-333(2000) [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]
- S. Akturk, M. Kimmel, P. O'Shea, and R. Trebino, "Measuring spatial chirp in ultrashort pulses using single-shot Frequency-Resolved Optical Gating," Opt. Express 11, 68-78(2003). [CrossRef] [PubMed]
- P. O'Shea, S. Akturk, M. Kimmel, and R. Trebino, "Practical issues in ultra-short-pulse measurements with ‘GRENOUILLE’," Appl. Phys. B 79, 683-691(2004). [CrossRef]
- S. Akturk, M. Kimmel, P. O'Shea, and R. Trebino, "Extremely simple device for measuring 20 fs pulses," Opt. Lett. 29, 1025-1027(2004). [CrossRef] [PubMed]
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