Improved FROG algorithm
Aravinth Ravichandran
Master Thesis May 2021
Department of Physics and Mathematics
University of Eastern Finland
Aravinth Ravichandran Improved FROG algorithm, 44 pages University of Eastern Finland
Master’s Degree Programme in Photonics Supervisors Dr. Matias Koivurova
Dr. Atri Halder
Abstract
Frequency–resolved optical gating [FROG] is one the most advanced ultrashort laser pulses measurement techniques today. It has evolved from conventional intensity autocorrelation techniques with frequency resolution step being added to the setup of autocorrelators. Introducing spectral resolution increases the information avail- able on the pulses to retrieve their phase and amplitude uniquely, through iterative Fourier transform FROG algorithm. Now, the scope of this thesis is to improve the robustness of the fundamental FROG algorithm with the power spectrum measured at high resolution as an additional constraint. Theoretical results shows that the improved FROG algorithm is simple, reliable, faster, capable of retrieving complex pulses, and noise insensitive. Despite the experimental errors, experimental results shows successful retrieval of central parts of pulses where most of the light energy is confined, with the improved FROG algorithm. Demonstrated with the second–
harmonic–generation [SHG] FROG, the improved algorithm is applicable to all the FROG variants.
Keywords: frequency–resolved optical gating; FROG; ultrashort laser pulses; phase retrieval; iterative Fourier transform algorithm; vanilla algorithm; power spectrum;
pulse designing; SLM; GRENOUILLE
Preface
From a light loving biologist to a light characterizing photonist, and everything in between.
This itsy–bitsy teeny–weeny evolutionary journey in my life has been made all possible, thanks to the Institute of Photonics and the University of Eastern Finland in Joensuu. I shall remain grateful to Prof. Dr. Jari Turunen, especially for this thesis topic. I am equally thankful to Dr. Matias Koivurova and Dr. Atri Halder for their support and guidance throughout my thesis.
This thesis is dedicated to all my well–wishers, friends, family and, to me.
Joensuu, the 17th of May 2021 Aravinth Ravichandran
Contents
1 Introduction 1
2 Ultrashort laser pulses 3
2.1 Mathematical definition . . . 3
2.2 Optical phases. . . 4
3 Pulse measurement problems 6 3.1 Autocorrelation . . . 6
3.2 Frequency–resolved optical gating – FROG . . . 9
4 Improved FROG algorithm 14 4.1 Additional constraint . . . 14
4.2 Simulations with perfect pulses . . . 16
4.3 Simulations with added noise . . . 20
4.4 Simulations and comparison with Vanilla algorithm . . . 24
5 Experimental realization 30 5.1 Pulse characterization setup . . . 30
5.2 Data fitting . . . 34
5.3 Experimental results . . . 35
6 Discussions and conclusion 38
References 41
Chapter I
Introduction
Light is life. It is central to the life sustaining process of photosynthesis [1] on our mother Earth. Light is magical. It helps us to perceive and cherish the world around through vision [2]. Light is fascinating and mind blowing, for example the rainbows, the lightnings, and the auroras. Light as we know it today is just a fraction of the electromagnetic spectrum [3]. The electromagnetic spectrum details the origin, characteristic properties, matter interaction, and practical significance of electromagnetic waves in it.
Ever since the advent of lasers in 1960, humankind has achieved unimaginable precision levels notably in manufacturing, metrology, military, and medicine [4,5].
Furthermore, they also paved the way for efficient storage of data, faster commu- nication, and even the Nobel winning ultra–high resolution imaging [4,6]. With the world sharply drifting towards nanotechnology, laser technologies drifts even be- yond and into the intriguing science of ultrafast optics and ultrashort optical pulses.
G´erard Mourou and Donna Strickland’s high–intensity ultrashort optical pulses have already bagged the Physics Nobel prize recently [7]. Their ground breaking inven- tion with ultrashort optical pulses have opened the floodgates for dwelling deep into the regime of strong–field physics and attosecond science [8].
There has been a heavy rely on the ultrashort laser pulses for their significant role in high–speed electronic and optoelectronic devices and systems characterization [9].
Quantum–world unique novel matter synthesis [9] and high–power ultrashort laser pulse induced thermal fusion [4,10–12] for electricity are one of its exciting appli- cations, indeed. Shapes of the ultrashort laser pulses are invaluable in microlithog- raphy for the manufacturing of electronic integrated circuits [9]. Also, control over
the shape of laser pulses could fasten the communication speed up to terabits per second [9]. Above all, precise knowledge of laser pulses and their properties improves the understanding of Physics of lasers [13].
For all these scientific advancements that the ultrashort laser pulses had offered and for all the scientific breakthroughs that are on offer with them, the ultrashort laser pulses are best studied and understood with the frequency–resolved optical gating [FROG] [13] technique invented by Rick Trebino and Daniel J. Kane in 1991.
Heart of the FROG technique lies in its pulse retrieving FROG algorithm. The scope of this thesis is to improve the robustness of the fundamental FROG algorithm with an additional constraint. For illustrative purpose, algorithmic demonstration is carried out on the second–harmonic–generation [SHG] FROG.
This thesis is built up in such a way that it introduces ultrashort laser pulses and optical phases in chapter II. This will be followed by the history and evolu- tion of ultrashort laser pulses measurement techniques in chapter III. In chapter IV, improved FROG algorithm is proposed and back–supported with theoretical simulations. Practical implementation of the improved algorithm in the laboratory and its associated results are detailed in chapter V. Finally, the results obtained are discussed and the thesis is concluded in chapter VI. MATLAB has been utilized as the tool for the purpose of improved algorithm implementation and demonstration.
Chapter II
Ultrashort laser pulses
This chapter introduces ultrashort laser pulses and it also defines the parameters that characterizes a laser pulse completely. It also sets up the basis of pulse designing which will later be utilized in the thesis for the testing of the improved FROG algorithm.
Optical pulses in the order of picoseconds and less are generally considered to be ultrashort. Ultrashort pulses are a short burst of electromagnetic radiation.
High resolution in space–time, higher bandwidth and intensity are its characteristic features. Ultrashort laser pulses are generated with mode locked lasers. They play a key role in the measurement of ultrafast events and they are one of the backbones of ultrafast optics. [14]
2.1 Mathematical definition
Mathematically, ultrashort laser pulses can be defined in the complex analytical signal form. The complex electric field of an ultrashort laser pulse in time domain is denoted by
E(t) = √︁
I(t) exp[−iϕ(t)], (2.1)
whereI(t) =|E(t)|2 represents the temporal intensity andϕ(t) represents the phase.
Similarly, the pulse’s complex electric field in the frequency domain is E(ω) = √︁
S(ω) exp[−iψ(ω)]. (2.2)
In Eq.(2.2),S(ω) = |E(ω)|2 represents the power spectrum or simply the spectrum, while ψ(ω) represents the spectral phase. The Fourier transform pairs relation be- tween E(t) and E(ω) is driven by
E(ω) =
∫︂ ∞
−∞
E(t) exp(−iωt) dt, (2.3)
and
E(t) = 1 2π
∫︂ ∞
−∞
E(ω) exp(iωt) dω. (2.4)
Eq.(2.2) can also be considered as a function of wavelength (λ) in the spectral domain. Eq.(2.1) and Eq.(2.2) completely characterizes the laser pulse, the former in the time domain, while the latter in the frequency domain or in the spectral domain. [13]
2.2 Optical phases
The terms ϕ(t) and ψ(λ) in Eq.(2.1) and in Eq.(2.2) are the optical phases. The effect of spectral phase on the temporal shape of the pulse can easily be understood from Fig. 2.1. It is realizable from Fig. 2.1 that the pulse remains unchanged in time, when the spectral phase is constant. Linear or first–order spectral phase causes the pulse to shift in time, while the spectral phase of the second–order is concerned with the temporal pulse spread. Third–order spectral phase causes the pulse to have side lobes in addition to the main lobe, while a random spectral phase makes the temporal pulse shape to be a random one. The very same effects can be observed on the spectrum due to temporal phase changes. [13]
Thus, controlling the spectral phase controls the pulse shape. Concept of pulse designing with spectral phase modulation have been utilized later in this thesis.
Figure 2.1: The effect of spectral phase, ψ(λ) on the temporal shape of the pulse. They are shown side by side. Solid blue line and solid brown line in the spectral domain plot represents the spectrum and the spectral phase. Solid blue line in the time domain plot is a pristine temporal pulse, while the dashed magenta line signifies the effect of spectral phase on the it.
Chapter III
Pulse measurement problems
This chapter briefs the historical evolution of ultrashort laser pulses measurement techniques.
Measuring a pulse in either of the domains is sufficient. Considering the frequency domain, spectrum can trivially be measured with a spectrometer. Then, measuring the spectral phase completes the pulse already. But, there exists almost no devices to measure optical phases. Iterative algorithms could be used to retrieve the spectral phase from the spectrum. Yet, for a given spectrum, there are nearly infinite spectral phases possible as the retrieval is one–dimensional. An attempt to measure the pulse in the time domain is the intensity autocorrelation. [13]
3.1 Autocorrelation
Pulse measurement history begins with the intensity autocorrelation [15–17]. Here, the pulse to be measured is splitted into two, variably delayed with respect to one another, and then recombined in a nonlinear crystal. It is that the output pulse energy from the nonlinear crystal is measured as a function of delay and referred to as the autocorrelation signal. Mathematically, it is
A(2)(∆t) =
∫︂ ∞
−∞
I(t)I(t−∆t) dt. (3.1)
In Eq.(3.1), I(t) and I(t−∆t) are the recombining pulses intensities with delay
∆t between them and A(2)(∆t) is the intensity autocorrelation signal or simply, the autocorrelation signal. A SHG autocorrelator is shown in Fig. 3.1 and SHG autocorrelation of some pulses are shown in Fig. 3.2.
SHG crystal
Lens Mirror Lens
Beam splitter Delay stage
Unknown pulse
I(t) I(t - t)
Detector A(2)(t)
Figure 3.1: SHG autocorrelator. Input unknown pulse is splitted into two, variably delayed with respect to one another and then recombined in a SHG crystal. SHG pulse energy is measured as a function of delay.
-5000 0 500
1 Pulse
-5000 0 500
1 Autocorrelation signal
-5000 0 500
1
-5000 0 500
0.5 1
I(t) [a.u.]
-5000 0 500
1
I(t) [a.u.]
-20000 0 2000
1
-2000 0 2000
0.5 1
-2000 0 2000
t [fs]
0 1
-2000 0 2000
t [fs]
0.5 1
-5000 0 500
1
Figure 3.2: Pulses and their SHG autocorrelation signal. Address it side–by–side.
Other nonlinear phenomenon such as the two–photon absorption and two–photon fluorescence can also be utilized to effect the autocorrelation [18,19]. There are a few more autocorrelators available involving polarization–gate [PG], self–diffraction [SD], and third–harmonic–generation [THG] autocorrelation techniques. [13]
Autocorrelation theorem is obtained by Fourier transforming Eq.(3.1) as
A(2)(ω) =|I(ω)|2, (3.2)
where I(ω) is the Fourier transform of I(t). From Eq.(3.2), it becomes clear that only I(t) can be retrieved from the autocorrelation signal, but not ϕ(t) which is the phase. Iteratively retrieving the phase from the autocorrelation signal is one–
dimensional as well, possessing almost infinitely possible phases for a given signal of the autocorrelation. Thus, much alike the spectrum, autocorrelation is insufficient as well in measuring the ultrashort laser pulses.
Moreover, even the pulse intensity cannot be uniquely determined with the help of autocorrelation, as different pulses shared similar autocorrelation profiles [20].
This can also be inferred from Fig. 3.2 that the autocorrelation signal is almost the same even when the pulses are clearly different and distinguishable from one another.
Properties of autocorrelation is that their signal always has its maximum at zero delay and they are always symmetrical even for non–symmetrical pulses. Also, the autocorrelation signal simply reduces to a coherent spike on a pedestal feature in the case of complex pulses [21]. In addition, this pulse measurement technique is highly noise sensitive and so the results are quite misleading in the presence of noise.
Despite the shortcomings, autocorrelation technique provides an exact measure of root–mean–squared [RMS] pulse width. With time, several technical improvements have been made to this autocorrelation pulse measurement technique. [13,22–26]
3.2 Frequency–resolved optical gating – FROG
Both the attempts to measure the pulse in the individual domains have failed, but never in the hybrid [27,28] time–frequency domain. Frequency–resolved optical gating [FROG] technique involving pulse measurement through self–gating makes excellent use of this hybrid domain indeed, and succeeds in completely measuring the ultrashort laser pulses. It has evolved from conventional autocorrelation with a trivial, yet an ingenious intermediate step. By spectrally resolving the pulse’s autocorrelation signal with a spectrometer, FROG [29–31] is born. Invented by Rick Trebino and Daniel J. Kane in 1991, this technique has become the golden standard for the measurement of ultrashort pulses. SHG FROG [32] is one of the simplest FROGs and it is shown in Fig. 3.3.
Signal Pulse SHG crystal
Lens
Lens Spectrometer
Mirror
Beam splitter Delay stage
Unknown pulse
Gate pulse
Probe pulse
Figure 3.3: SHG FROG setup. An unknown pulse to be measured is split into a probe pulse and a gate pulse. These pulses are variably delayed with respect to each other and then recombined in a SHG crystal. Later, the output signal pulse from the SHG crystal is spectrally resolved with the help of a spectrometer, resulting in a FROG trace which simply is the signal pulse’s spectrum as a function of delay.
It is even the most famous of all the invented FROGs as well. It is highly sensitive and it can even measure pulses with energy as low as 1 pJ and a few femtoseconds [fs]
long. It has the best signal–to–noise ratio amongst the FROGs and it is also capable of measuring complex pulses [33]. Their FROG traces are always symmetrical about zero delay, and mathematically, a SHG FROG trace is given by
ISHG(∆t, ω) =
⃓
⃓
⃓
⃓
∫︂ ∞
−∞
E(∆t, t) exp(−iωt) dt
⃓
⃓
⃓
⃓
2
, (3.3)
where
E(∆t, t) =E(t)E(t−∆t), (3.4)
and
E(∆t) =E(t). (3.5)
In Eq.(3.4),E(t) represents the electric field of the probe pulse,E(t−∆t) represents the electric field of the gate pulse, and E(∆t, t) represents the electric field of the signal pulse. SHG FROG traces of pulses featured in Fig. 3.2 are given in Fig. 3.4.
Figure 3.4: SHG FROG traces of pulses featured in Fig. 3.2. Figures has to addressed in a side–by–side manner.
FROG traces are unique for a given pulse and this can also be realized from the presented SHG FROG traces in Fig. 3.4. Symmetrical nature of SHG FROG traces can also be seen in Fig. 3.4. Besides, much alike the vastly available autocorrelators, there exists an equally available FROGs, including but not restricted to PG FROG, SD FROG, THG FROG, transient–grating [TG] FROG, and cascaded [CC] FROG.
Each and every FROG are unique and comes up with their own advantages and disadvantages. [13]
It is the phase retrieval associated with the FROG trace that makes the FROG technique successful in determining the properties of the pulses. Unlike the one–
dimensional iterative phase retrievals in spectrum and autocorrelation, FROGs it- erative phase retrieval is two–dimensional. With some additionally available con- straints, this iterative two–dimensional phase retrieval process possess a singular solution. Heart of the FROG lies in its phase retrieving iterative Fourier transform algorithm [29].
Generically, fundamental FROG algorithm, also referred to as the vanilla algo- rithm works as depicted in Fig. 3.5.
E(t) / E( ) E( t, )
E'( t, ) E'(t) / E'( )
Start
Generate trace
Input measured amplitude
Recover field Update
guess
Figure 3.5: Working of vanilla FROG algorithm.
The algorithm starts by guessing E(t). In the case of SHG FROG, the algorithm makes use of Eq.(3.4) to generate E(∆t, t). E(∆t, t) is then Fourier transformed with respect to time t to generate E(∆t, ω). Further, the magnitude of resulted E(∆t, ω) is replaced with the magnitude of the measured SHG FROG trace while keeping its phase intact as in Eq.(3.6) as to generate E′(∆t, ω).
E′(∆t, ω) =√︁
ISHG(∆t, ω) exp[iϕ], (3.6) where
ϕ = arg [E(∆t, ω)], (3.7)
is the argument of the retrieved trace. E′(∆t, ω) is then inverse Fourier transformed with respect to ω to generate E′(∆t, t). From E′(∆t, t), a new guess for E(t) can be generated by a simple integration over the delay ∆t accordingly to Eq.(3.8)
E′(t) =
∫︂ ∞
−∞
E′(∆t, t) d∆t, (3.8)
and Eq.(3.9)
E(t) =E′(t). (3.9)
The newly obtainedE(t) will be then utilized to make a newE(∆t, t). Next iteration starts and this procedure goes on until convergence. Convergence is estimated with FROG error [G(k)] defined by
G(k) =
⌜
⃓
⃓
⎷ 1 N2
N
∑︂
i,j=1
|ISHG(∆ti, ωj)−ISHG(k) (∆ti, ωj)|2. (3.10) In Eq.(3.10), the summation stretches over the whole extent of the delay and fre- quency coordinates (∆ti, ωj), and the measured SHG FROG trace [ISHG] is normal- ized to the same intensity as the k–th iteration of the retrieved trace [ISHG(k) ]. Size of traces is N. For the vanilla FROG algorithm to converge, two constraints have to be fulfilled and it is shown as a Venn diagram in Fig. 3.6. Guessed pulse is projected onto each of the constraints as in Fig. 3.6 until the real pulse is obtained.
The intersection area between the constraints contains the desired pulse result. In the absence of errors, the intersection area shrinks to a unique pulse. [13]
In addition, FROG method of measurement of ultrashort laser pulses is unique for its self–consistency data checkpoints. These checkpoints collectively called the FROG marginals [34] are one–dimensional curves obtained by integrating a FROG trace over one of its coordinates. For a SHG FROG, the marginals are
M(∆t) =
∫︂
ISHG(∆t, ω) dω, (3.11)
and
M(ω) =
∫︂
ISHG(∆t, ω) d∆t. (3.12)
Eq.(3.11) is the delay marginal which is the integral of SHG FROG trace over all of its frequencies. Eq.(3.12) is the frequency marginal which is the integral of SHG FROG trace over all of its delays. Delay marginal is considered equivalent to be the pulse’s intensity autocorrelation signal while the frequency marginal is considered equivalent to be the pulse’s spectrum autoconvolution. Thus, delay marginal and frequency marginal can be used to check if the FROG trace is correctly measured.
This can be confirmed by comparing respective marginals with their identities as above–mentioned. With FROG marginals, not only the FROG measurement errors are spotted, but they can also be corrected. The correction can simply be applied by multiplying the measured FROG trace with the ratio of autoconvolution of pulse’s spectrum and the frequency marginal. [13]
Solutions that satisfy mathematical constraints
Solutions that satisfy measurement constraints Desired result
Starting guess
Figure 3.6: Convergence properties of the vanilla FROG algorithm.
Chapter IV
Improved FROG algorithm
In this chapter, improved FROG algorithm is proposed and it is back–supported with numerical simulations.
The fundamental vanilla FROG algorithm is quick and simple, but it often does not converge for complex pulses. It is sensitive to noise as well. Also, the algorithm is fairly reliable in the retrieval of very simple pulses alone. Several improvements have been made to this vanilla FROG algorithm [35]. Recently, a more reliable and faster FROG algorithm was demonstrated by incorporating the power spectrum retrieved from the FROG trace in the process of pulse retrieval [36].
4.1 Additional constraint
Practically, a FROG trace does not contains the best estimate of the power spectrum.
Moreover, a power spectrum is trivially measurable and at very high resolution with a simple spectrometer. It is this power spectrum measured at high resolution [Sm(ω)]
that could be used as an additional constraint to the fundamental FROG algorithm.
By constraining so, the reliability and the speed of the fundamental FROG algorithm could be improved. This is achievable since the measured power spectrum could be used to update the guess for the pulse in the algorithm accordingly to Eq.(4.1)
E(ω) =√︁
Sm(ω) exp[iψ], (4.1)
where
ψ = arg [E′(ω)] (4.2)
is the argument [phase] of the retrieved field. Importantly, because of the applied constraint, a situation is attained where all the guesses made for the pulse are strictly within this additional constraint data, thus inheriting reliability and speed to the algorithm. Otherwise, the improved FROG algorithm works as exactly as in section 3.2. Working model of the proposed improved FROG algorithm is shown in Fig. 4.1 and its convergence properties in the likes of a Venn diagram is given in Fig. 4.2.
E(t) / E( ) E( t, )
E'( t, ) E'(t) / E'( )
Start
Generate trace
Input measured amplitude
Recover field update with
measured spectrum
Figure 4.1: Improved FROG algorithm working model.
Solutions that satisfy mathematical constraints
Solutions that satisfy measurement constraints Additional
data constraint
Figure 4.2: Convergence properties of the improved FROG algorithm.
4.2 Simulations with perfect pulses
A Gaussian spectrum centered at 792 nm with a spectral width of 10 nm was simu- lated. To this, a sinusoidal (amplitude = 0.5, frequency = 46 Hz), a hysteresis like, and a random phase spectral phase was added to generate three different spectral fields according to Eq.(2.2). These spectral fields were then Fourier transformed us- ing Eq.(2.4) to generate pulse fields corresponding to a trident pulse1, a Bloch pulse2, and a random pulse3. It is these simulated pulse fields that were used in simulating their respective SHG FROG traces based on Eq.(3.3). Given these simulated –traces and –spectrum, the improved FROG algorithm was then used to retrieve the pulses corresponding to the given traces and spectrum. Such retrieval results are shown in Figs. 4.3, 4.4, and 4.5.
The pulses retrieval was set to end automatically once achieved a root–mean–
square [RMS] difference of less than 0.5% between the simulated and the retrieved traces within a maximum of 100 iteration cycles. The trace RMS difference was defined using Eq.(3.10). Also defined was the RMS difference between the simulated and the retrieved spectrum called to be the spectral RMS difference [Srms] according to Eq.(4.3).
Srms =
√︃ 1 N
∑︂|S(λ)s−S(λ)r|2, (4.3) whereS(λ)s was the simulated spectrum andS(λ)r was the spectrum retrieved from k–th iteration cycle of the algorithm.
It can be seen from Figs. 4.3, 4.4, and 4.5 that the improved FROG algorithm is able to retrieve the simulated –trident pulse, –Bloch pulse, and the –random pulse perfectly, that too in less than 35 iteration cycles. Spectral RMS difference is observed to be less than 1% in all the three performed pulse retrievals. In the case of retrieval of the trident pulse and the Bloch pulse, the algorithm was started with a flat spectral phase. For the random pulse retrieval, the algorithm was initiated with a random spectral phase.
1pulse with three separate peaks
2a Gaussian pulse with two peaks
Figure 4.3: Improved FROG algorithm retrieval of the trident pulse. Simu- lated and retrieved traces are shown in parallel with RMS difference between them along with the number of iteration cycles it took for the algorithm to achieve less than 0.5% trace RMS difference. Simulated and retrieved spectrum are shown in solid blue line and in dashed magenta line with RMS difference between them in the spectral domain plot. In the same plot, retrieved spectral phase is shown in solid brown line. Simulated and retrieved pulse are shown in solid blue line and in dashed magenta line in the time domain plot along with the retrieved phase in solid brown line. Trace RMS difference as a function of number of iteration cycles is shown in the convergence plot. Traces size: N by N pixels. N = 512.
Figure 4.4: Improved FROG algorithm retrieval of the Bloch pulse. Simulated and retrieved traces are shown in parallel with RMS difference between them along with the number of iteration cycles it took for the algorithm to achieve less than 0.5% trace RMS difference. Simulated and retrieved spectrum are shown in solid blue line and in dashed magenta line with RMS difference between them in the spectral domain plot. In the same plot, retrieved spectral phase is shown in solid brown line. Simulated and retrieved pulse are shown in solid blue line and in dashed magenta line in the time domain plot along with the retrieved phase in solid brown line. Trace RMS difference as a function of number of iteration cycles is shown in the convergence plot. Traces size: N by N pixels. N = 512.
Figure 4.5: Improved FROG algorithm retrieval of the random pulse. Simu- lated and retrieved traces are shown in parallel with RMS difference between them along with the number of iteration cycles it took for the algorithm to achieve less than 0.5% trace RMS difference. Simulated and retrieved spectrum are shown in solid blue line and in dashed magenta line with RMS difference between them in the spectral domain plot. In the same plot, retrieved spectral phase is shown in solid brown line. Simulated and retrieved pulse are shown in solid blue line and in dashed magenta line in the time domain plot along with the retrieved phase in solid brown line. Trace RMS difference as a function of number of iteration cycles is shown in the convergence plot. Traces size: N by N pixels. N = 512.
4.3 Simulations with added noise
It was to challenge the algorithm more, uniformly distributed multiplicative noise was added to the simulated traces according to Eq.(4.4),
In(∆t, ω) = |E(∆t, ω) (1 +rand(N, N) (√
n))|2. (4.4)
In Eq.(4.4), n was the user defined noise level and In(∆t, ω) was the noised simu- lated trace. User defined noise level was set to 100%. Given the spectrum along with the noised trace, subsequent pulse retrieval was performed for the previously demonstrated –trident pulse, –Bloch pulse , and –random pulse, adapting the same algorithm definitions and end conditions narrated as in section 4.2. Such pulse retrieval results are given in Figs. 4.6,4.7, and 4.8.
As it can be seen from the Figs. 4.6, 4.7, and 4.8 that the improved algorithm is able to retrieve the pulses seamlessly and flawlessly even when the user defined additive noise level is set to 100%. Also, it is observed from the convergence plots in Figs. 4.6,4.7, and4.8 that it took almost 30 iteration cycles for the algorithm to retrieve the noised –trident pulse and –random pulse, and almost 60 iteration cycles in the retrieval of noised Bloch pulse. Noise addition indeed prolonged the pulses retrieval by a few more iteration cycles and this becomes evident in comparison with the convergence plots of the respective pulses in Figs. 4.3,4.4, and4.5. Surprisingly, yet again in comparison, addition of 100% uniformly distributed multiplicative noise to the simulated traces just led to almost 1% increase of trace RMS difference. The algorithm indeed ran for all the allotted number of 100 iterations cycles for all the three pulses retrieval as the trace RMS difference did not go less than the set 0.5%
during the process.
Figure 4.6: Improved FROG algorithm retrieval of 100% uniformly distributed multiplicative noise added trident pulse. Trace retrieved at the end of the 100th iteration cycle and the simulated noisy trace are shown in parallel with RMS difference between them. Simulated and retrieved spectrum are shown in solid blue line and in dashed magenta line with RMS difference between them in the spectral domain plot. In the same plot, retrieved spectral phase is shown in solid brown line. Simulated and retrieved pulse are shown in solid blue line and in dashed magenta line in the time domain plot along with the retrieved phase in solid brown line. Trace RMS difference as a function of number of iteration cycles is shown in the convergence plot. Traces size: N by N pixels. N = 512.
Figure 4.7: Improved FROG algorithm retrieval of 100% uniformly distributed multiplicative noise added Bloch pulse. Trace retrieved at the end of the 100th iteration cycle and the simulated noisy trace are shown in parallel with RMS difference between them. Simulated and retrieved spectrum are shown in solid blue line and in dashed magenta line with RMS difference between them in the spectral domain plot. In the same plot, retrieved spectral phase is shown in solid brown line. Simulated and retrieved pulse are shown in solid blue line and in dashed magenta line in the time domain plot along with the retrieved phase in solid brown line. Trace RMS difference as a function of number of iteration cycles is shown in the convergence plot. Traces size: N by N pixels.
N = 512.
Figure 4.8: Improved FROG algorithm retrieval of 100% uniformly distributed multiplicative noise added random pulse. Trace retrieved at the end of the 100th iteration cycle and the simulated noisy trace are shown in parallel with RMS difference between them. Simulated and retrieved spectrum are shown in solid blue line and in dashed magenta line with RMS difference between them in the spectral domain plot. In the same plot, retrieved spectral phase is shown in solid brown line. Simulated and retrieved pulse are shown in solid blue line and in dashed magenta line in the time domain plot along with the retrieved phase in solid brown line. Trace RMS difference as a function of number of iteration cycles is shown in the convergence plot. Traces size: N by N pixels. N = 512.
4.4 Simulations and comparison with Vanilla algorithm
Special features of improved FROG algorithm could be well appreciated by simply comparing it alongside its mother algorithm, which is the vanilla FROG algorithm.
The MATLAB code for vanilla algorithm is available online [37]. That code was downloaded and defined for trace and spectral RMS difference as in section 4.2.
The algorithms were compared adapting the same simulation end conditions used in section4.2. Given the simulated traces of trident pulse, Bloch pulse, and random pulse as in Figs. 4.3, 4.4, and 4.5 as the input, vanilla algorithm was then expected to retrieve the respective pulses given. Such vanilla algorithm pulse retrieval results are given in Figs. 4.9, 4.10, and 4.11.
From the time domain plots and the traces in Figs. 4.9, 4.10, and 4.11, it becomes clear that the vanilla algorithm successfully retrieved the pulses, indeed.
On comparing vanilla algorithm retrieved results in Figs. 4.9, 4.10, and 4.11 with the improved FROG algorithm retrieved results in Figs. 4.3, 4.4, and 4.5, it also becomes strikingly clear that the improved FROG algorithm outperformed vanilla algorithm in the following aspects.
Firstly, the lower trace RMS difference associated with the improved algorithm.
This value signifies that the retrieved traces are numerically much closer to the simulated traces. Since the trace contains the pulse information, numerically closer traces gives out a strong impression that the pulses retrieved from them should also be numerically much closer to the simulated pulses. This unique trait must be attributed towards the usage of the spectrum as an additional constraint in the improved algorithm. Secondly, it just takes fewer number of iteration cycles for the improved algorithm to converge that too to a trace RMS difference of less than 0.5% is set. Thirdly and perhaps the most significant of them all would be the retrieved spectrum from the improved algorithm. The retrieved spectrum always maintained the simulated spectrum profile and it is exactly this feature that makes the improved FROG algorithm more reliable. Misleading nature of vanilla algorithm is perfectly revealed in the spectral domain plot of Fig. 4.10, where it retrieved a spectrum different from the simulated one that too with similar looking traces even in the theoretical simulations. All these arguments made on the improved FROG algorithm remains valid even on comparing its noise added pulse retrieval results from Figs. 4.6, 4.7, and 4.8 with vanilla retrieved results from Figs. 4.9, 4.10, and
4.11. One more valid argument that could be made on the improved algorithm is that it always exactly converged to the random pulse and its trace which has been under demonstration throughout this chapter [Figs. 4.5 and4.8]. While the same is not true with the vanilla algorithm. At few of the retrieval instances of the random pulse with the vanilla algorithm, convergence was never achieved and one of this non–convergence instances is show in Fig. 4.12. There, the non–convergence is easily noticeable from the non–similar traces and pulses. Moreover, the expected spectrum and the retrieved spectrum in their spectral domain plot do not match as well.
Comparing between the vanilla algorithm based random pulse retrieval results from Figs. 4.11and4.12, it also gives out an impression that the matching of the expected and the retrieved spectrum could be a significant event during the convergence of the vanilla algorithm. Understanding this argument from another perspective, this is exactly the aim of improved FROG algorithm, that the retrieved and the simulated spectrum stays the same at the end of the retrieval process, thereby making the FROG algorithm more reliable. However, this spectrum matching argument with regard to the vanilla algorithm and its associated convergence needs to be confirmed more.
To summarize, thus the pulse retrieving fundamental FROG algorithm becomes more reliable with the improved FROG algorithm which incorporates spectrum as an additional constraint in the pulse retrieval process. Also, the improved algorithm remarkably lessens the number of iteration cycles required for the pulse retrieval, as all the pulse guesses are within this additional constraint data. It also works very well in retrieving complex pulses and does that so even when the noise is present.
In addition, it retrieves numerically more closer pulses and traces [accurate] as well.
This improved FROG algorithm should converge for any pulses in principle.
Figure 4.9: Vanilla algorithm retrieval of trident pulse corresponding to the simulated trace in Fig. 4.3 and its convergence. Trace retrieved after the 100th iteration cycle and the input given trace are shown alongside with RMS difference between them. Expected and retrieved spectrum are shown in solid blue line and in dashed magenta line with RMS difference between them in the spectral domain plot. In the same plot, retrieved spectral phase is shown in solid brown line. Expected and retrieved pulse are shown in solid blue line and in dashed magenta line in the time domain plot along with the retrieved phase in solid brown line. Trace RMS difference as a function of number of iteration cycles is shown in the convergence plot. Traces size: N by N pixels.
N = 512.
Figure 4.10: Vanilla algorithm retrieval of Bloch pulse corresponding to the simulated trace in Fig. 4.4 and its convergence. Trace retrieved after the 100th iteration cycle and the input given trace are shown alongside with RMS difference between them. Expected and retrieved spectrum are shown in solid blue line and in dashed magenta line with RMS difference between them in the spectral domain plot. In the same plot, retrieved spectral phase is shown in solid brown line. Expected and retrieved pulse are shown in solid blue line and in dashed magenta line in the time domain plot along with the retrieved phase in solid brown line. Trace RMS difference as a function of number of iteration cycles is shown in the convergence plot. Traces size: N by N pixels.
N = 512.
Figure 4.11: Vanilla algorithm retrieval of random pulse corresponding to the simulated trace in Fig. 4.5 and its convergence. Trace retrieved after the 100th iteration cycle and the input given trace are shown alongside with RMS difference between them. Expected and retrieved spectrum are shown in solid blue line and in dashed magenta line with RMS difference between them in the spectral domain plot. In the same plot, retrieved spectral phase is shown in solid brown line. Expected and retrieved pulse are shown in solid blue line and in dashed magenta line in the time domain plot along with the retrieved phase in solid brown line. Trace RMS difference as a function of number of iteration cycles is shown in the convergence plot. Traces size: N by N pixels.
N = 512.
Figure 4.12: Vanilla algorithm retrieval of random pulse corresponding to the simulated trace in Fig. 4.5and its non–convergence. Trace retrieved after the 100th iteration cycle and the input given trace are shown alongside with RMS difference between them. Expected and retrieved spectrum are shown in solid blue line and in dashed magenta line with RMS difference between them in the spectral domain plot. In the same plot, retrieved spectral phase is shown in solid brown line. Expected and retrieved pulse are shown in solid blue line and in dashed magenta line in the time domain plot along with the retrieved phase in solid brown line. Trace RMS difference as a function of number of iteration cycles is shown in the convergence plot. Traces size: N by N pixels.
N = 512.
Chapter V
Experimental realization
This chapter details the testing of the improved FROG algorithm in the laboratory.
It also explains the steps involved in the processing of the measured data.
5.1 Pulse characterization setup
Experimental realization of the improved FROG algorithm required the power spec- trum and the FROG trace of a pulse to be measured.
In the laboratory, an Integra–C 3.5 Ti:sapphire near infrared pulsed laser [38]
with Gaussian pulses of FWHM of less than 130 fs at 1 kHz frequency was available.
The central wavelength of the laser was 792 nm and the pulse energy was about 3.5 mJ/s. The laser output was linearly polarized along the vertical direction and the laser beam measured approximately 8 mm in diameter.
Power spectrum of this laser was measured with a grating–camera setup as shown in Fig. 5.1.
To test the improved algorithm practicality, priori known pulses were designed utilizing spectral phase modulation technique as demonstrated in section 2.2. In the laboratory, this spectral phase modulation was achieved with the available Hamamatsu photonics X13267–02 liquid crystal on silicon spatial light modulator [SLM] [39] device. This device working on the principle of birefringence, was a phase only–reflective type SLM capable of modulating input light’s phase by more than 2π radians. This SLM also accepts designed spectral phase as a MATLAB image of size 600 by 800 pixels, wherein the numerical values 0–0.8672 corresponds to a spectral phase of 0–2π radians.
Laser
Mirror
Polarizer
Gaussian filter
Grating
Lens
Camera
/4 plate
Figure 5.1: Power spectrum measurement setup. Laser beam from the source first passes through a λ/4 plate, polarizer, and a Gaussian filter. The beam diameter after passing through the Gaussian filter measures 4 mm. The beam will then pass through a reflective grating [in here, shown with transmission grating for easier illustration] and thus the power spectrum is measured by the camera.
In the laboratory, also available was the GRENOUILLE [GRating–Eliminated No–nonsense Observation of Ultrafast Incident Laser Light E–fields] [40]. It is an ultrasimple SHG FROG device co–invented by Rick Trebino. It inherits none of the optical components from the traditional SHG FROG setup. Apparently, they are replaced with simple smart optics and elements. GRENOUILLE coming with up pre–installed FROG pulse retrieval algorithm software characterizes the input light pulse completely, in a single shot manner, that too in real–time. It also saves the retrieval data and even displays them.
The experimental setup utilized in measuring the FROG trace of a pulse is shown in Fig. 5.2.
Laser
Mirror
Polarizer
Gaussian filter
Mirror
Grating
Lens
SLM GRENOUILLE
/4 plate
Designed spectra
l phas e
image
Mirror
Figure 5.2: FROG trace measurement setup. Laser beam from the source first passes through a λ/4 plate, polarizer, and a Gaussian filter. The beam diameter after passing through the Gaussian filter measures 4 mm. The beam will then pass through a reflective grating [in here, shown with transmission grating for easier illustration] thereby setting up the SLM with the spectrum for the spectral phase modulation to take place. Also, to the SLM, spectral phase of a pulse of interest is given in the form of a spectral phase image.
Then, spectrally modulated light from the SLM after passing through the grating becomes the pulse of interest. This pulse is then be measured by the GRENOUILLE. FROG trace of this measured pulse is then taken from the GRENOUILLE.
Before made the actual pulse measurements, the pristine pulses from the laser were
quadratic spectral phase image for the SLM in the laboratory during the measure- ments, so that a constant spectral phase profile was reached and observed in the GRENOUILLE, utilizing the experimental setup in Fig. 5.2. A constant spectral phase leads to the generation of bandwidth–limited pulses [13]. It was on these bandwidth–limited laser pulses, subsequent pulse designing was carried out by addi- tion of this found optimized converse quadratic spectral phase image along with the spectral phase image originally designed and meant for the generation of the pulses of interest.
A trident pulse and a Bloch pulse were designed and measured using the setup as in Fig. 5.2. Spectral phase images used in their designing along with their respective GRENOUILLE measured FROG traces are given in Fig. 5.3.
Measured trace of Bloch pulse
-500 0 500
t [fs]
380
390
SH() [nm] 400
Measured trace of trident pulse
-500 0 500
t [fs]
385 390 395 400 405
SH() [nm]
SLM phase for trident pulse designing
200 400 600 800 Pixels
100 200 300 400 500 600
Pixels
0 0.2 0.4 0.6 0.8 SLM phase for Bloch pulse designing
200 400 600 800 Pixels
100 200 300 400 500 600
Pixels
0 0.2 0.4 0.6 0.8
Figure 5.3: Originally designed spectral phase images with their pulse’s GRENOUILLE measured FROG traces. Traces size: 1200 by 1600 pixels.
5.2 Data fitting
GRENOUILLE measured data of trident and Bloch pulse were loaded and processed in MATLAB in such a way that they fit into the computational window of the im- proved FROG algorithm. It was done so by fixing first the computational window to have traces of size 64 by 64 pixels and to have a total time delay of 3000 fs. Imple- menting time–frequency relationship and centering the second–harmonic wavelength axis [SH(λ)] at 396 nm [exactly one half of the central wavelength of the laser used, and due to SHG type of FROG], SH(λ) axis precisely corresponding to the total delay was calculated. This calculated SH(λ) axis indeed provided the wavelength range for the measured spectrum to be considered from, in the algorithm.
With the axes for the measured –trace and the –spectrum being fixed, the mea- sured –FROG trace and the –spectrum were accordingly cropped and interpolated to ultimately fit into the improved FROG algorithm’s computational window.
In addition to data fitting into the computation window, improved algorithm also required optimal scaling of the measured trace in the SH(λ) axis, so that it becomes comparable with the retrieved trace along that axis. This scaling step was primarily needed because of the squeezed nature of measured trace along the SH(λ) axis. This can easily be realized by looking at the measured FROG traces in Fig.
5.3. The measured FROG trace should equally be spread in both of its axes for proper convergence of the algorithm [13]. Besides, this scaling of the measured trace was vital for the visual appreciation of convergence from the traces, as well. Scaling was accomplished by trial and error method involving random removal of set of lines of pixels along the SH(λ) axis of the measured traces in Fig.5.3, followed by data re–sampling to its original size.
It was all these processed measured data of the designed –trident pulse and – Bloch pulse that were given as the input to the improved FROG algorithm for the respective pulses retrieval.
5.3 Experimental results
With the adapted algorithm definitions and end conditions as in section 4.2, pulses retrieval were performed until an optimum scaling as briefed in section 5.2 was found for the given measured traces. Scaling success was quantified with visual interpretation of the traces and the RMS difference between them.
After several tries and trials, a point beyond which the trace RMS difference do not lessen, irrespective of the scaling of the measured traces, was reached. Improved FROG algorithm pulse retrieval results attained at that critical scaling point for the measured –Bloch pulse and –trident are given in Figs. 5.4 and 5.5.
Figure 5.4: Critical scaling point reached improved FROG algorithm retrieval of measured Bloch pulse from the laboratory. Critical scaled GRENOUILLE measured trace and improved FROG algorithm retrieved trace after the 100th iteration cycle are shown side by side with RMS difference between them.
Measured power spectrum is shown in solid blue line and the improved FROG algorithm retrieved spectrum is shown in dashed magenta line in the spectral domain plot with RMS difference between them. In the same plot, improved FROG algorithm retrieved spectral phase is shown in solid brown line and GRENOUILLE retrieved spectrum is shown in dashed blue line. In the time domain plot, improved FROG algorithm retrieved pulse is shown in dashed magenta line and GRENOUILLE retrieved pulse is shown in dashed blue line.
In the same plot, improved FROG algorithm retrieved phase is shown in solid brown line. Trace RMS difference as a function number of iteration cycles is shown in the convergence plot. Traces size: 64 by 64 pixels.
Figure 5.5: Critical scaling point reached improved FROG algorithm retrieval of measured trident pulse from the laboratory. Critical scaled GRENOUILLE measured trace and improved FROG algorithm retrieved trace after the 100th iteration cycle are shown side by side with RMS difference between them.
Measured power spectrum is shown in solid blue line and the improved FROG algorithm retrieved spectrum is shown in dashed magenta line in the spectral domain plot with RMS difference between them. In the same plot, improved FROG algorithm retrieved spectral phase is shown in solid brown line and GRENOUILLE retrieved spectrum is shown in dashed blue line. In the time domain plot, improved FROG algorithm retrieved pulse is shown in dashed magenta line and GRENOUILLE retrieved pulse is shown in dashed blue line.
In the same plot, improved FROG algorithm retrieved phase is shown in solid brown line. Trace RMS difference as a function number of iteration cycles is shown in the convergence plot. Traces size: 64 by 64 pixels.
Chapter VI
Discussions and conclusion
In this chapter, obtained experimental results are discussed in detail and the thesis is concluded based on the observations and inferences made from the results.
It can be inferred from Figs. 5.4 and 5.5 that the improved FROG algorithm retrieved traces are noticeably looking different from the GRENOUILLE measured traces, which is why the RMS difference between them is high and the algorithm ran for all the allotted 100 iteration cycles. Ideally, the GRENOUILLE measured traces of trident pulse and Bloch pulse should have been symmetrical about the zero delay, as the GRENOUILLE is a SHG FROG device [40]. Characteristic feature of SHG FROG traces are their symmetrical nature about the zero delay [13]. This confers the existence of possible experimental errors during the measurement of FROG traces of pulses, even when the FROG marginals from the measured traces were fine. Moreover, it can also be inferred from the GRENOUILLE measured FROG traces that there is a possible defect with the SHG crystal inside the device.
This is being inferred from the presence of no second–harmonic [SH] signal region, especially on the left side from zero delay in the measured traces from Figs. 5.4and 5.5. Also, in the measured traces, the left side from zero delay is poorly measured in comparison with its counter–parting right side. A simple solution to improve the left side of these measured traces would be mirroring their right side. By doing so, the measured traces would also become symmetrical. The main reason for non–
similar looking traces in Figs. 5.4 and 5.5 must be attributed to the process of scaling of the measured traces. Upon achieving ideal scaling with fine–tuning, the algorithm would have retrieved similar looking traces. This in turn would have made the pulses retrieval results more appealing. Also, scaling of the measured traces is
random. This makes data processing of the measured traces a non–trivial task. In this regard, a simple and smart data processing step should be devised for the scaling of measured traces to test the improved FROG algorithm in a very good manner.
Also, since that the power spectrum is measured and that the spectral phase is designed to test the improved FROG algorithm through priori designed pulses, it is intriguing to go onto make an expected pulse using first the Eq.(2.2) followed by Eq.(2.4) from which an expected trace could also be made using Eq.(3.3). By mathematically constructing this expected trace, it would become easier to probe the experimental errors involved in the measurement of FROG traces of pulses with GRENOUILLE, with more certainty. Essentially, this mathematical construction of the expected –traces and –pulses would considerably improve the reliability with which the improved FROG algorithm is being tested.
Now, despite the experimental errors with GRENOUILLE, the improved FROG algorithm, still successfully retrieved the central parts of the pulses, where most of the pulse energy is contained. This is being inferred from central overlapping of the improved FROG algorithm retrieved pulses with the GRENOUILLE retrieved pulses from the time domain plots of Figs. 5.4 and 5.5. Plotting of mathematically constructed expected pulse on these very same plots would add–in more reliability on the algorithm testing and as well as on the GRENOUILLE.
It can also be inferred from the spectral domain plots in Figs. 5.4 and 5.5 that the improved FROG algorithm retrieved spectrum shares a similar profile with the measured spectrum, though the RMS difference between them is high in both the pulses retrieval. From the same plots, it can also be seen that the GRENOUILLE retrieved spectrum varies in its profile in comparison with the measured spectrum.
Besides, improved FROG algorithm always retrieved a better spectrum than what GRENOUILLE retrieved, and in comparison with the measured spectrum. This in turn makes the improved FROG algorithm fare better against the algorithm used in GRENOUILLE [13]. Higher spectral RMS difference could be attributed to the iterative nature of the algorithm used even when the measured spectrum is directly being employed.
In summary, theoretical results shows that the FROG pulse retrieval algorithm becomes more reliable with the improved FROG algorithm which incorporates power spectrum as an additional constraint in the process of pulse retrieval. It also shows that the improved FROG algorithm retrieves more accurate pulses that too in lesser iteration cycles. The improved FROG algorithm also retrieves complex pulses and works well, even in the presence of noise. Whereas the experimental results shows that despite the experimental errors involved, the improved FROG algorithm suc- cessfully retrieved the central parts of the pulses tested. Though demonstrated with SHG FROG, the concept is still applicable to the whole lot of the FROG family.
Thus, the improved FROG algorithm is impressive indeed in improving the robust- ness of the existing fundamental FROG pulse retrieval algorithm with the power spectrum as an additional constraint. Excellent phase guess feature could be an add–on to this improved FROG algorithm.
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