Recent progress toward real-time measurement of ultrashort laser pulses

Abstract

Frequency-resolved optical gating (FROG) is a technique that
produces a spectrogram of an ultrashort laser pulse optically. While a
great deal of information about the pulse can be gleaned from its FROG
trace, often it is desirable to obtain of the pulse information
immediately, in real time. Quantitative information about the guise is
not readily obtainable from its spectrogram without the use of a
two-dimensional phase retrieval algorithm. While current algorithms are
quite robust, retrieval of all the pulse information can be slow. In
this paper, I describe a recently developed FROG trace inversion
algorithm called Principal Component Generalized Projects that is fast,
robust, and can invert FROG traces in real time. A femtosecond
oscilloscope based on second-harmonic generation FROG is also described
that uses this new algorithm to rapidly (up to 2.3 Hz) and continuously
display the intensity and phase of ultrashort laser pulses

Full text

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 35, NO. 4, APRIL 1999 421
Recent Progress Toward Real-Time Measurement
of Ultrashort Laser Pulses
Daniel J. Kane
(Invited Paper)
Abstract—Frequency-resolved optical gating (FROG) is a tech-
nique that produces a spectrogram of an ultrashort laser pulse
optically. While a great deal of information about the pulse
can be gleaned from its FROG trace, often it is desirable to
obtain all of the pulse information immediately, in real time.
Quantitative information about the pulse is not readily obtainable
from its spectrogram without the use of a two-dimensional phase
retrieval algorithm. While current algorithms are quite robust,
retrieval of all the pulse information can be slow. In this paper, I
describe a recently developed FROG trace inversion algorithm
called Principal Component Generalized Projects that is fast,
robust, and can invert FROG traces in real time. A femtosecond
oscilloscope based on second-harmonic generation FROG is also
described that uses this new algorithm to rapidly (up to 2.3 Hz)
and continuously display the intensity and phase of ultrashort
laser pulses.
Index Terms— Phase retrieval, pulse measurement, ultrafast
lasers, ultrashort pulses.
I. INTRODUCTION
F
REQUENCY-RESOLVED optical gating (FROG) is used
to characterize ultrashort laser pulses [1]–[12]. It optically
obtains a spectrogram of an input pulse by interacting one
or more pulses in a nonlinear medium to form a gate that
interacts with the input pulse. The interaction forms a signal
pulse which is spectrally resolved and recorded as a function
of delay between the input pulse and the gate. The spectrogram
(FROG trace) is a plot of signal intensity versus frequency and
time. The target information, the temporal and spectral profiles
of the input pulse (intensity and phase), can be obtained from
the FROG trace using two-dimensional (2-D) phase-retrieval
methods [4], [13], [14].
FROG is experimentally simple and data acquisition can be
rapid: less than 70 ms using a video camera and frame grab-
ber. The resulting spectrogram provides immediate qualitative
information about the pulse. Quantitative pulse characteris-
tics require up to a few minutes to obtain (depending on
the required resolution) because of the iterative nature of
the phase-retrieval calculation [4], [15]–[18]. Thus, FROG’s
usefulness as a real-time diagnostic for ultrashort laser pulses
depends on the speed and robustness of the phase-retrieval
algorithm used. In this paper, I discuss the development of a
new algorithm to obtain quantitative pulse characteristics, in
Manuscript received October 6, 1998; revised December 23, 1998. The
work of D. J. Kane was supported by the National Science Foundation under
Grant III-9361715 and Grant DMI-9 801 116.
The author is with Southwest Sciences, Inc., Santa Fe, NM 87505 USA.
Publisher Item Identifier S 0018-9197(99)02548-8.
real time, from experimental FROG traces. The paper begins
with a brief summary of inversion algorithm development. The
discussion continues with the description of a new inversion
algorithm, called the Principal Component Generalized Projec-
tions Algorithm (PCGPA) [17], [18], that is very fast for some
common FROG geometries. Later, I combine the PCGPA with
data acquisition in a multishot second-harmonic generation
(SHG) FROG [19], [20] device to develop a femtosecond
oscilloscope that demonstrates real-time pulse measurement,
displaying the intensity and phase of the extracted pulse at
rates up to 2.3 Hz [18].
II. FROG I
NVERSION ALGORITHMS
The first step in all inversion algorithms is to construct
a spectrogram mathematically that mimics a physical FROG
device (see Fig. 1). An input pulse can be represented by the
equation
(1)
where
and are the time-dependent intensity and
phase, respectively, and
is the carrier frequency. Upon
entering the FROG device, the pulse is split into two identical
pulses via a beam splitter. The identical pulses are combined in
a nonlinear material producing a signal with the mathematical
form
(2)
where
is referred to as the probe, and is the gate
function that converts the pulse into the gate, which depends on
the nonlinear interaction used. For the FROG device depicted
in Fig. 1,
because an SHG crystal is
used and the second harmonic is collected. Another nonlinear
interaction commonly used in FROG devices is the optical
Kerr effect in the polarization-gate (PG) geometry [2], [3]. In
that case,
.
A spectrometer spectrally resolves the signal; that is mim-
icked in the algorithm via a Fourier transformation into the
frequency domain. A detector, such as a CCD array, obtains
the FROG trace by recording spectral intensity of the signal
at each time delay. These data can be represented as the
magnitude squared of the Fourier transform of
(3)
0018–9197/99$10.00  1999 IEEE

422 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 35, NO. 4, APRIL 1999
Fig. 1. Schematic of an SHG FROG device. A beamsplitter splits the input
into probe and gate beams. The two beams are focused into an SHG
crystal. The spectrum of the second harmonic is collected as a function of
delay.
Fig. 2. All FROG trace inversion algorithms work by iterating between two
constraints related by a transformation with an inverse. Paramount to the
algorithm’s performance is how it obtains the estimate of
for the next
iteration.
is a real quantity; therefore, it has no direct phase
information. The goal of the FROG inversion algorithm is to
determine the phase by solving the equation
(4)
for
which is a complex function of unity magnitude.
Thus, (2) and (3) define the two constraints common to all
FROG algorithms that must be satisfied [4], [21]. Equation
(2) is called the physical constraint and is used in the FROG
algorithms both to obtain the next guess for
and to
construct the new signal field. It is applied in the time domain.
The result from (3), the FROG trace, is the intensity constraint
which is applied in the frequency domain (4). The goal of the
algorithm is to minimize the difference between the measured
FROG trace and the FROG trace calculated from the current
pulse
[see (5)] [4].
Fig. 2 shows the general form of FROG trace inversion
algorithms. An initial guess is provided for
to
get the algorithm started [4].
a guess for
is calculated and Fourier transformed into
is replaced by the
square root of the measured FROG trace,
.
The next guess for
is then calculated from
. Consequently, the phase-retrieval
problem looks much like a recursive equation with
variables (where is the number of time points and
frequency points).
Fig. 3. Figure showing the ideal operation of a FROG inversion algorithm.
Once the new estimate for the is obtained, a new spec-
trogram is constructed. The process is repeated (see Fig. 2)
until the spectrogram error
(equivalent to the FROG trace
error) reaches an acceptable minimum
(5)
where
represents the rms error per element of the
spectrogram,
is the current iteration of the
spectrogram,
is the measured spectrogram, and
and are the th frequency and th delay in the frequency
and delay vectors, respectively [4].
The various FROG algorithms differ in how
is cal-
culated from
[4], [15]–[18], [21].
If convergence time was of no concern, the first try at an
algorithm might be a simulated annealing algorithm where the
next guess for
is obtained randomly from the previous
guess for
. If only we had forever! A better approach
is to produce the next guess from the previous guess in a
systematic way, but how?
Ideally, after each iteration of the algorithm, a slightly
better estimate for the phase of the spectrogram is obtained
until convergence (Fig. 3). The original FROG inversion al-
gorithm integrated the time domain FROG trace,
with respect to the time delay, to obtain subsequent guesses
for
(so-called “vanilla” or “basic” algorithm) [4], [15],
[21]. The integration effectively reduces the gate function
to a constant, yielding
where is the integration
constant and
is the next guess for the electric field.
While fast, this algorithm stagnates easily and fails to invert
spectrograms of double pulses [4], [15], [21]. In an attempt
to overcome stagnation problems and increase robustness, the
vanilla algorithm was used to provide an initial guess to a brute
force minimization of the rms difference between the retrieved
FROG trace and the experimental FROG trace [15]. While this
method is robust, converging in most cases, it is very slow.
A major advance in the FROG inversion algorithm came
with the development of the generalized projections algorithm
[16], [19], [22]–[24]. First, it virtually guarantees that the error
always decreases for each iteration [16], [24]. Second, the

KANE: RECENT PROGRESS TOWARD REAL-TIME MEASUREMENT OF ULTRASHORT LASER PULSES 423
generalized projections algorithm is very robust [16], [17],
[21]. Third, it is reasonably fast, being much faster than
brute force minimization [16]. Last, it converges well even
in the presence of noise. A complete discussion, along with
derivatives used in the minimization, appears in a review
article by Trebino et al. [21].
Like the previous algorithms, generalized projections works
by alternating between two (or more) sets,
and [21].
(For FROG trace inversion,
is the set of all ’s
satisfying the nonlinear material response, and
is the set of
all complex functions with a magnitude
.) Unlike
the previous algorithms, generalized projects finds the next
guess by insuring that the distance between
and is
minimized for each iteration of the algorithm. DeLong et al.
developed a generalized projections algorithm for FROG by
using a minimization algorithm to find the
that minimizes
the Euclidian distance between the signal field constructed
from
and the signal field that satisfies the intensity
constraint
. That is, the
following equation:
(6)
is minimized with respect to
to obtain the next estimate
of
and is the function that converts into the
gate function [16], [19], [21].
The algorithm developed by DeLong et al. directly follows
the definition of generalized projections [22]–[24]. The im-
plementation is very powerful; it can be used for any FROG
geometry and can include material response [25], but it is too
slow for real-time inversion of FROG traces [18]. For common
FROG geometries, such as SHG and PG, a new generalized
projections algorithm, called Principal Component Generalized
Projections (PCGP) [17], has recently been developed that
does not require a minimization step, increasing the iteration
rate by nearly a factor of two [18]. Because the PCGPA
code is compact, it easily fits into inexpensive digital signal
processing boards, allowing simultaneous data acquisition
and FROG trace inversion. Using such a scheme, this new
algorithm has been used to invert FROG traces in real time
and has been demonstrated in a femtosecond oscilloscope that
can continuously and indefinitely characterize ultrashort laser
pulses in real time [18]. Sections III–V discuss the derivation
of this new algorithm.
III. P
RINCIPAL COMPONENT GENERALIZED
PROJECTIONS ALGORITHM
In FROG, there is always an assumed relationship be-
tween the probe and the gate. However, there is another, less
common, but more general case where the gate is entirely
independent of the probe. This has been called TREEFROG
(Twin Retrieval of Excitation Electric fields FROG) by the
more acronymous among us [26]. I shall refer to this general
case as blind-FROG because it makes no a priori assumptions
about the relationships between the probe and the gate [17].
At first, this may seem obtuse, but it allows the development
of a generalized projections algorithm that does not require
a minimization step, ultimately speeding algorithm execution
and simplifying programming. (This becomes important later
when the inversion algorithm is placed entirely on a digital
signal processing board.)
Thus, the blind-FROG spectrogram,
is
(7)
where the gate is represented as
and the probe
as
. The function that produces is not
required. A single time slice of the blind-FROG trace is the
intensity spectrum of the product of these two functions where
the gate is delayed relative to the probe by
. The complete
spectrogram is obtained when the gate is scanned in time
across the probe,
.
Equation (1) is the magnitude squared of the Fourier trans-
form of the product
with respect to . Virtually
all practical data collection methods rely on discretizing
and
. Suppose and are sampled at given values of
with a constant spacing of . Then and can be
thought of as vectors of length
whose elements sample
and at discrete times
(8)
For simplicity, the vectors are written as
(9)
The outer product of
and is (Fig. 4)
(10)
This matrix will be referred to as the outer product form.
The outer product contains all of the points required to
construct the time domain FROG trace because it contains
all of the interactions between the pulse and gate for the
discrete delay times. Consequently, a one-to-one mapping of
the elements of the outer product can transform the outer
product into the time domain of the FROG trace. This is the
key to the PCGP algorithm. Because the mapping is one-to-
one, it is invertible; transformations can be made from the
outer product form to the time domain FROG trace and vice

424 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 35, NO. 4, APRIL 1999
(a) (b)
Fig. 4. The different steps in the PCGPA. (a) SHG and (b) PG are shown.
The top image plots show the outer product. The next image results from
the row rotation depicted in (11). By rearranging the columns, the correctly
oriented time-domain FROG trace can be constructed. Fourier transforming
the columns produces the FROG traces shown in the bottom image plots.
versa. This transformation can be accomplished by rotating the
elements of the rows in the outer product to the left by the row
number minus one. Applying this transformation, we obtain
(11), shown at the bottom of the page. The
column is
the first column, where
is the time delay in increments of
a point-by-point multiplication of the probe by the gate with
no time shift between them. The next column is the
column where the gate is delayed relative to the probe by one
resolution element,
. The gate appears to be shifted “up” by
one resolution element with the first element wrapped around
to the other end of the vector. Column manipulation places the
most negative
on the left and the most positive on the right.
Thus, (11) is the time domain of the spectrogram formed by
the multiplication of the probe and gate functions; a discrete
version of the product
. The columns
are constant in
(delay) while the rows are constant in
(time). This gives exactly the same result as calculating the
time-domain FROG trace directly by shifting the gate in time
and multiplying the shifted gate by the probe. All I am doing
is insuring is that there is a reversible way to move between
the outer product form and the time-domain FROG trace. By
Fourier transforming each column, the Fourier transform of
is obtained as a function of . The final
step of taking the magnitude of the complex result produces
the FROG trace.
IV. PCGPA I
NVERSION
It is easy to imagine an infinite number of complex images
that have the same magnitude as the spectrogram we wish to
invert; however, there is only one image that can be formed by
the outer product of a single pair of nontrivial vectors that has
the same magnitude as the spectrogram to be inverted. In order
to find the proper vector pair, the phase of the spectrogram
must be determined using a 2-D phase-retrieval algorithm.
Like all FROG trace inversion algorithms, the PCGPA is
started using Gaussian pulses with random phase for the initial
guess for
. The initial gate pulse is derived from
according to the FROG geometry used. A spectrogram is
constructed and its magnitude is replaced by the square root
of the magnitude of the experimentally obtained spectrogram.
The result is converted to the time-domain spectrogram (11)
using an inverse Fourier transform by column (see Fig. 4).
Next, the time-domain spectrogram is converted to the outer
product form (10) by reversing the steps used to construct the
time domain spectrogram. If the intensity and phase of the
spectrogram are correct, this matrix (the outer product form
matrix) is a true outer product and has a rank of one. That is, it
would have one and only one nonzero eigenvalue and one right
eigenvector and one left eigenvector. The right eigenvector,
the probe, spans the range of the outer product matrix. The
complex conjugate of the eigenvector of the transpose of the
outer product matrix (left eigenvector) is the gate [24].
The outer product form matrix produced by the initial guess,
however, is not rank one and has several eigenvectors. It
will probably have (for an
FROG trace) right
eigenvectors and
left eigenvectors (eigenvectors of the
transpose): instead of describing of a single line in
space,
the matrix represents an ellipsoid in
space. The best
(11)

KANE: RECENT PROGRESS TOWARD REAL-TIME MEASUREMENT OF ULTRASHORT LASER PULSES 425
next guess may actually be a superposition of two or more
different but linearly independent eigenvectors, requiring an
optimization such as minimization of the FROG trace error to
find the correct superposition.
Fortunately, minimization is not required. Suppose we de-
compose the outer product form matrix
into three matrices
such that
(12)
where
and are orthogonal square matrices and
is a square diagonal matrix. Thus, the matrix the outer
product form, is decomposed into a superposition of outer
products between “probe” vectors (columns of
) and “gate”
vectors (rows of
). The diagonal values in (the only
nonzero elements of
) determine the relative weights of each
outer product and, therefore, how much each outer product
contributes to matrix
. If we keep the outer product pair
with the largest weighting factor, or principal component, for
the next iteration of the algorithm, we minimize the function
(13)
where
is the outer product form matrix, is the
probe vector,
is the gate vector, and is the error
[27]. (This is the definition of a projection and is similar
to, although not identical to, the projection found in the
generalized projections algorithm developed by DeLong et al.
[16], [19], [21].)
How can all this be accomplished? One elegant, but com-
putationally intensive, means to find the principal vector
pair is to use a singular value decomposition (SVD) to
decompose
into and directly [24], [27]. This
approach is convenient because many commercially available
mathematical libraries contain routines to compute SVD’s.
Another way to find the principal vector pair with much
less computation than an SVD is to reduce the SVD step to
simple low-overhead and fast matrix-vector multiples [28]. For
real-time applications, this is the best approach [18].
Rather than finding the eigenvectors of
the outer product
form matrix, and constructing an orthonormal basis from these
vectors, an SVD finds the eigenvectors of
(columns of
and (columns of which are orthonormal [24],
[27]. If the columns of
are written as and the columns
of
are written as then they satisfy the equations
(14)
where
’s are the eigenvalues, or “weights,” and the su-
perscript
is the transpose operator. may be constructed
by
(15)
where
and are provided by the SVD, but we only
need the
and corresponding to the largest or
Fig. 5. Schematic of the PCGP algorithm. Transformation from the outer
product to the time domain FROG trace (and vice versa) may be accomplished
via simple permutations (rotations) of each row.
the principal eigenvectors. Suppose we multiply an arbitrary
nonzero vector
by . Then
(16)
where
is the eigenvector of the eigenvalues, and
a set of constants. can be thought of as an operator that
maps
onto a superposition of eigenvectors. The process can
be repeated resulting in
. Multiplying
by
gives
(17)
As
becomes large, the largest eigenvalue dominates the
sum so that
. This method is called
the power method [28]. After a few iterations, a very close
approximation to the principal eigenvector (the eigenvector
with the greatest eigenvalue) is obtained. Consequently, the
next guess for the pulse can be obtained by multiplying the
previous guess for the pulse by
. The next guess for the
gate can be obtained by multiplying the previous guess for the
gate by
. (For polarization-gate FROG, the absolute value
of the result for the gate is taken.) While better approximations
for the eigenvectors may be obtained by using these operators
several times per iteration, once per iteration is adequate in
practice [18].
Practically, the power method implementation of the PCGP
algorithm (Fig. 5) is very fast and quite robust. Indeed, the
power method implementation can loop at nearly 20 itera-
tions/s on a 60 MFLOPS digital signal processor or greater
than 30 iterations/s on a 233-MHz Pentium II. Good approx-
imations for the pulse usually occur in about 40 iterations
[18].
V. C
ONVERSION OF PCGPA TO A FROG
I
NVERSION ALGORITHM
The PCGPA, as discussed above, is a blind-FROG al-
gorithm. That is, the probe and the gate are completely
independent. The only nonlinear interaction assumed is the
multiplication of the probe by a gate. How the gate is con-
structed is of no concern. As a result, some ambiguities
can occur. Even though these ambiguities are usually minor,

426 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 35, NO. 4, APRIL 1999
TABLE I
C
OMPARISON OF SHG FROG INVERSION ALGORITHM
Algorithm Type
Random Noise
Test: Percent
Convergence
Random Chirp
Test: Percent
Convergence
Multipulse Test:
Percent
Convergence
SHG Vanilla 15 (4) 45 (32) 64 (3)
Femtosoft SHG
FROG
56 (48) 80 (52) 92 (0)
Fast SHG PCGPA 78 (77) 79 (70) 95 (10)
Results of a test designed to determine robustness of FROG inversion
algorithms. All of the test groups are synthetic. Models for the test groups
are discussed in Kane [18[. The percentages given are for a percent of pulses
retrieved with an rms FROG trace error of less than 2
10 in 100 iterations.
This was deemed to be the lowest experimental error that can, in practice, be
achieved and more aptly defines the usefulness of the algorithm. Percentages
given in parentheses are for strict convergence with an rms error of less than
10
in 100 iterations. The ultimate convergence rates (allowing the algorithm
to continue past 100 iterations until stagnation) for the Femtosoft SHG FROG
inversion algorithm were 60%, 80%, and 32% for the random noise test,
the random chirp test, and the multipulse test, respectively. The ultimate
convergence rates of the SHG FROG PCPG algorithm was not determined.
they can produce erroneous results if ignored (see Appendix
A) [17], [26]. Spectral constraints can facilitate inversion of
FROG spectrograms using the PCGPA [17], [26]. This method
has been used extensively to invert experimental FROG traces
and blind-FROG traces [29], [30]. Often, however, a spectrum
is not available; consequently, we would rather not be required
to obtain a spectrum of the pulse in addition to its spectrogram.
The conversion of PCGPA to a FROG algorithm may be
accomplished by summing the outer product of the probe and
the gate with the outer product of the probe constructed from
the gate and the gate constructed from the probe [18]. How the
gate is constructed from the probe and vice versa is determined
from the nonlinear interaction. In the case of SHG FROG, for
example, the probe is equal to the gate; thus, the outer product
becomes
(18)
forming the FROG trace from the sum of two outer products.
Because only the principal outer product pair is used for the
next estimate of the electric field, the two outer products are
forced to be equal. The only way the outer products can be
equal is if probe
gate.
This type of FROG algorithm works very well for SHG
FROG (see Appendix B). Table I compares the PCGP-based
SHG FROG algorithm to the commercially available Fem-
tosoft SHG FROG program [19], [21], and to the “Basic
FROG” or “Vanilla” algorithm [4], [21]. The three algorithms
were tested to failure using three synthetically constructed
test sets. From the test results, it can be determined that the
generalized projections-based algorithms are clearly superior
to the “Vanilla” algorithm. In the first test, the random noise
test, which determines the overall robustness of the algorithms,
the SHG PCGP algorithm performed best. In the other two
tests, the PCGP SHG algorithm compares favorably to the
Femtosoft algorithm [18].
SHG FROG is a special case, however; (18) is valid only for
SHG FROG and must be modified for other FROG geometries.
As defined in (2),
is the function that produces the gate from
the probe
its inverse, denoted produces the probe
from the gate. Thus, the gate
and the probe
. Rather than using only the outer product of
and to produce the next time-domain FROG
trace, the sum of the outer products of
and
is used so that the outer product on
the next iteration is given by
(19)
where
is the sum of the two outer products for the th
iteration.
Equation (19) allows PCGPA to be used with any FROG
geometries where
exists. In the PG FROG, however, the
inverse of the gate function does not exist. As a result, a
pseudo-inverse must be constructed from the square root of the
gate intensity and the phase of the pulse. Because the square
root can cause small fluctuations in the wings of the gate,
producing artifacts in the next guess for the pulse, instabilities
may occur in the algorithm. This can be remedied by applying
the square root to only well-defined portions of the gate. Where
the gate is not well defined (i.e., the intensity is near zero),
the intensity (and phase) of the pulse is used. To increase the
robustness of the PG algorithm, the pseudo-inverse constraint
is applied on alternate iterations. The pseudo-inverse method
works well for polarization-gate (PG) FROG on the synthetic
test sets, converging to experimental error for 90% of the test
pulses, but it has not been tested with experimental data. At
this time, the pseudo-inverse method does not appear to work
well for self-diffraction (SD) FROG, however.
VI. E
XPERIMENTAL—THE FEMTOSECOND OSCILLOSCOPE
The development of the PCGPA can facilitate the inversion
of FROG spectrograms in real time [18]. However, building
a femtosecond oscilloscope requires more than just a fast
inversion algorithm. The data acquisition must be completely
integrated with the inversion algorithm. This is accomplished
in the demonstration described here by integrating the data ac-
quisition and the inversion engine with a home-built multishot
SHG FROG device. The data acquisition and inversion engine
utilizes two commercially available digital signal processing
(DSP) boards (Fig. 6). This device successfully demonstrates
the inversion of experimental FROG traces in real time and
can display the inverted pulses (from a 64
64 FROG trace)
at a rate of 1.25 Hz, or one every 0.8 s, and 2.3-Hz inversion
rates were possible for a 32
32 array.
Using a zero-dispersion pulse stretcher-compressor [31] to
vary the pulse dispersion independent from the Ti:sapphire
oscillator allows extensive testing of the femtosecond oscil-
loscope. By translating the lens, dispersion in the beam can
be changed enough to more than triple the pulsewidth. The
femtosecond oscilloscope can easily track these changes. Also,
portions of the spectrum can be blocked to shape the pulse
before being sent to the FROG device (Figs. 7 and 8).
Example data obtained using the femtosecond oscilloscope
are shown in Fig. 7. The FROG trace shown in Fig. 7(a) was
produced by blocking a portion of the pulse spectrum at the
Fourier plane of the stretcher-compressor. Also show in Fig. 7
are the retrieved pulse, the retrieved phase, and an example of

KANE: RECENT PROGRESS TOWARD REAL-TIME MEASUREMENT OF ULTRASHORT LASER PULSES 427
Fig. 6. Schematic of the femtosecond oscilloscope. A multishot SHG FROG
device with a rapid scanning delay line acts as the front end. The detector
is an EG&G 512 element diode array. The array is read by the data DSP
which also controls the delay line. After each spectrum is read, the delay
line is incremented by
until a complete spectrogram is obtained. The
spectrogram is then resampled, filtered, and sent to the host computer. The
host displays the spectrogram and sends it to the inversion DSP. After about
15 iterations, the pulse and gate are read by the host and displayed. This
device could fully characterize pulses at a rate of 1.25 Hz for 64
64 FROG
traces and 2.3 Hz for 32
32 FROG traces.
(a) (b)
(c) (d)
Fig. 7. Data taken using the femtosecond oscilloscope depicted in Fig. 5. (a)
The FROG trace: this pulse was retrieved using the PCGPA as the inversion
engine implemented on the DSP card. In only one second (20 iterations), the
algorithm converged to a FROG trace error of less than 0.5% for a 64
64
FROG trace. Also important to the operation of the femtosecond oscilloscope
is the stability of the algorithm; results did not change significantly even after
thousands of iterations. (b) Pulse. (c) Algorithm/DPS processing time. (d)
Pulse phase.
the performance of the DSP/PCGPA combination. After only
one second of computational time, the algorithm ran for 20
iterations on the 64
64 FROG trace, converging to a FROG
trace error of less than 0.5%.
The SHG FROG device splits the input beam into two
identical beams by a beam splitter [18]. One beam is sent into
a manual delay line used to fine tune the delay between the
two beams so that the FROG trace is centered along the time
axis for proper operation of the PCGP algorithm. The other
beam is sent into a fast scanning delay line based on a General
Scanning LT 1000 Z (linear) scanner [18[ allowing the delay
to be controlled by a voltage
2-mm delay/V). The resulting
beams are about 8 mm apart and focused by a 250-mm focal
length lens into a 200-
m-thick BBO crystal. The spectrum of
the second harmonic is measured via a 1/4 m spectrograph and
a 512–element EG&G Reticon diode array controlled by the
EG&G demonstration board. The resulting electronic signal is
filtered by a SRS 560 low-noise differential amplifier before
being digitized at 100 kHz (5 ms exposure) by the 16-bit A/D
converters on the data collection DSP board. After the diode
array is read, the translation stage is set to the next delay via
a D/A on the DSP board. Sixty-four spectra are obtained for
the 64
64 FROG trace and 32 for the 32 32 FROG trace.
In addition to digitizing the diode array readout, the data
collection DSP board also prepares the raw data for input
into the algorithm by resampling the signal vector from the
512-element diode array down to 64 pixels using a 15-
element finite impulse response digital filter. The coefficients
are chosen to remove all frequencies higher than Nyquist for
the resampled vector. After filtering, the background from
electronics offset and scattered light is subtracted.
The host computer (166 MHZ Pentium) controlled both DSP
boards which are each based on a single Texas Instruments
TMS320C32 floating point DSP (Fig. 6) via a host program
that polls the data acquisition DSP board for a new spectro-
gram. When ready, the host reads the spectrogram, frees the
data DSP board to read another spectrogram, and displays the
spectrogram (Fig. 8). The host then reads the new pulse and
gate from the inversion engine DSP board running the SHG
FROG PCGPA. The new spectrogram is sent to the inversion
engine board. The initial guess used by the algorithm in the
inversion DSP for the new spectrogram is the pulse retrieved
from the previous spectrogram. The reason behind this step is
the assumption that the average pulse will not change too much
over one second, allowing real-time updates to occur. This is
an important part of the femtosecond oscilloscope. For small
pulse changes, the algorithm will track continuously, which
is usually the case when adjusting a stretcher-compressor, for
example. However, for a step function change in the pulse,
such as blocking the input beam momentarily, the algorithm
can require 2 s to track the change (for a 64
64 FROG trace).
VII. C
ONCLUSIONS
While the PCGPA is naturally a blind-FROG algorithm,
it has been successfully adapted to the inversion of FROG
spectrograms by averaging the outer product of the pulse
and gate with the outer product of the pulse constructed

428 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 35, NO. 4, APRIL 1999
(a) (b)
Fig. 8. Femtosecond oscilloscope display. (a) The FROG trace resulting from a wire blocking out portions of the pulse spectrum at the Fourier plane
of a stretcher-compressor. (b) Both the pulse intensity and spectrum (spectral intensity of the retrieved pulse). Notice that the center portion of the
spectrum is missing and ringing of the pulse is clearly visible. To prevent the variable offset in the retrieved phase from changing the plot scaling
between updates, the derivative of the phase is displayed.
from the gate with the gate constructed from the pulse. Best
suited for SHG FROG, this algorithm is the most robust
algorithm available for the inversion of SHG FROG traces.
By using a pseudo-inverse, PCGP can also be used to invert
PG FROG spectrograms. While not quite as robust as other PG
FROG algorithms, it is still a viable alternative when speed,
simplicity, or compactness is of concern.
The main hindrance to real-time pulse measurement has
been the iterative algorithm required to invert the FROG traces.
Even with the fast processors available today, the computa-
tional requirements for obtaining the FROG trace, preprocess-
ing the FROG trace, and inverting the FROG trace in real time
are too great for most desktop computers. By the addition of
an inexpensive but separate DSP, independent of the operating
system, real-time pulse measurement can be a reality. Because
of the limited memory space on these DSP boards, the compact
vector-based fast PCGPA is ideal for this application.
When on a DSP board, the PCGPA using the power method
can run at nearly 20 iterations/s which is more than enough
to track small changes in the input pulse. For SHG FROG,
the PCGPA operates about two times faster than the current
generalized projections algorithm while being as robust as the
commercially available compound algorithm [21]. The con-
structed femtosecond oscilloscope obtains SHG FROG traces,
updates a spectrogram display, and provides the intensity and
phase of the pulse in real time with an update of 0.8 s or
1.25 Hz for a 64
64 array (0.43 s or 2.3 Hz for a 32 32
array). Currently under development is a real time femtosecond
oscilloscope based on a single-shot FROG device, a video
CCD camera, and a frame grabber.
A
PPENDIX A
B
LIND-FROG PITFALLS,EFFECTIVE AMBIGUITIES
While ambiguities are not a problem when inverting FROG
traces, blind-FROG inversions can have ambiguities [17], [26].
The first set of ambiguities is the order of the vector pair.
For example, one ambiguity in polarization-gate blind-FROG
occurs when there are no phase distortions present in the probe;
the blind-FROG trace does not change when the probe and
gate are interchanged. As a result, any blind-FROG algorithm
may converge with the probe and gate reversed. Minor phase
distortions in the probe can prevent this from occurring.
Of greater concern are small variations in the spectrogram
that can sometimes produce relatively large fluctuations in the
probe and gate vectors; this usually only occurs when the probe
and gate are similar in shape and duration. This results in
an effective ambiguity in polarization-gate blind-FROG. I call
this an effective ambiguity because it is not an ambiguity of
the technique, but rather manifests itself only when there are
noise and/or distortions on the spectrogram. Ideally, the set
containing all the spectrograms that have the same intensity
as the measured spectrogram and the set containing all of
the spectrograms that can be formed from the outer product
of two vectors intersect at only one point (Fig. 1). When
this occurs, we say the solution is unique. However, when
noise and distortions are present on the spectrogram, the two
sets no longer intersect at a point, but rather they intersect
within a region or, more likely, do not intersect at all. The
topology of the two sets indicates how successful the search
for a solution will be. If one or both of the sets are pointed

KANE: RECENT PROGRESS TOWARD REAL-TIME MEASUREMENT OF ULTRASHORT LASER PULSES 429
near the intersection point, ambiguities will not pose any
concern. However, if the region near the intersection, or closest
approach, is broad and flat on both sets, small changes in the
error can result in relatively large changes in the probe and
gate. One example is when a blind-FROG trace is produced
by a Gaussian probe pulse with a FWHM of 100 fs and a
Gaussian gate (the actual gate, not the gate pulse) with the
same FWHM. Such a trace will differ from a blind-FROG
trace produced by a Gaussian probe pulse with a FWHM of
105 fs and a Gaussian gate with a FWHM of 95 fs by an
of only 0.000 38. Subtle differences that are present can
be obscured by noise. On the other hand, if the probe and
gate have different shapes and very different widths, effective
ambiguities do not appear to be a problem.
In FROG, as opposed to blind-FROG, because of the a pri-
ori knowledge of the relationship between the probe and gate,
the width ambiguity is not a problem. These ambiguities may
be resolved when using a blind-FROG inversion algorithm
such as PCGPA by the addition of a spectral constraint on
either the probe or the gate [17], [26].
A
PPENDIX B
S
AMPLE MATLAB PROGRAM LISTING
’’ ’’
’
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-
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“ ”
“ ”
“ ”

430 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 35, NO. 4, APRIL 1999
“ ”
“ ”
MATLAB code for the inversion of SHG FROG traces.
This program will run without modifications under MATLAB
running on any platform. It returns the time reversed pulse. For
SHG FROG, this does not matter because of the ambiguity in
the direction of time. If this program is to be converted to PG
FROG, then this must be kept in mind.
A
CKNOWLEDGMENT
The author would like to thank D. Bomse for his helpful
suggestions.
R
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