Two-photon imaging of soliton dynamics Łukasz A. Sterczewski1 and Jarosław Sotor1 1Faculty of Electronics Photonics and Microsystems Wroclaw University of Science and Technology Wyb.

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Two-photon imaging of soliton dynamics

Łukasz A. Sterczewski1,∗, and Jarosław Sotor1

1Faculty of Electronics, Photonics and Microsystems, Wroclaw University of Science and Technology, Wyb.

Wyspianskiego 27, 50-370 Wroclaw, Poland

∗e-mail: lukasz.sterczewski@pwr.edu.pl

Optical solitary waves (solitons) that interact in a non-

linear system can bind and form a structure similar to a

molecule. The rich dynamics of this process have created

a demand for rapid spectral characterization to deepen the

understanding of soliton physics with many practical impli-

cations. Here, we demonstrate stroboscopic, two-photon

imaging of soliton molecules (SM) with completely unsyn-

chronized lasers, where the wavelength and bandwidth

constraints are considerably eased compared to conven-

tional imaging techniques. Two-photon detection enables

the probe and tested oscillator to operate at completely

different wavelengths, which permits mature near-infrared

laser technology to be leveraged for rapid SM studies of

emerging long-wavelength laser sources. As a demonstra-

tion, using a 1550 nm probe laser we image the behavior

of soliton singlets across the 1800–2100 nm range, and

capture the rich dynamics of evolving multiatomic SM. This

technique may prove to be an essential, easy-to-implement

diagnostic tool for detecting the presence of loosely-bound

SM, which often remain unnoticed due to instrumental

resolution or bandwidth limitations.

Since the discovery in hydrodynamic systems1, solitary

waves also known as solitons have expanded into diverse

areas of science due to the unique way they analogize mat-

ter and waves. Arguably, a signiﬁcant fraction of soliton

research has been centered around nonlinear optical sys-

tems such as ﬁbers2or microvavity resonators3,4. This is

because optical solitons do not spread out during propaga-

tion and exhibit robustness against perturbations; there-

fore they frame the core concept in optical pulse genera-

tion. Solitons also have the striking ability to form a stable

bond between pairs or groups referred to as SMs5,6. The

binding force7,8 depends on the atomic spacing τ(tem-

poral separation, TS), while the intramolecular phase ∆ϕ

mostly governs the attraction/repulsion eﬀect. Even more

sophisticated physical systems like molecular complexes9

or molecular crystals10 can form via SM collisions. Despite

advances in mathematical modeling11, the understanding

of these complex inter-soliton interactions still appears to

be in infancy.

The rich landscape of nonlinear dynamics and numer-

ous analogies to condensed-matter physics fuel intensive

research in this ﬁeld with much focus on studying the

evolution dynamics and transient behavior of SM forma-

tion12,13. It stems from the SM application potential to

optical memories, buﬀers14, or telecommunication to sur-

pass the limitation of classical binary coding schemes15,16.

In such scenarios, one would like to tailor the SM spac-

ing and phase on demand17,18,19,20 rather than rely on its

uncontrolled organization, which necessitates a thorough

characterization of transient short-lived SM states.

SM characterization techniques diﬀer signiﬁcantly in

obtainable scan rates. Time-averaged (second-scale, Hz-

rate) SM studies are performed with a ﬁrst-order inten-

sity autocorrelator (IAC) along with an optical spectrum

analyzer (OSA)21, which are suitable mostly for steady-

state phenomena like stable, tightly-bound SM. The high-

est scan rates (shot-to-shot, sub-GHz) are oﬀered by the

celebrated Dispersive Fourier Transformation (DFT) tech-

nique22,23, which temporally stretches a laser pulse in a

linear dispersive medium (such as optical ﬁber) to map the

time domain to the frequency domain on a pulse-by-pulse

basis12,13,24,18,25. Unfortunately, beyond the 1–2 µm win-

dow, ﬁber losses may render this technique impractical.

Also the spectral resolution of the two techniques (typi-

cally ∼GHz) may be insuﬃcient to probe loosely-bound

SMs (TS on the order of ns). A much simpler variant of

this method – direct analysis of laser pulses on an oscil-

loscope without dispersive stretching – completely ignores

the molecular phase and fails to resolve solitons spaced by

ps due to electrical bandwidth limitations.

To ﬁll a niche between the two timescales, recent works

have proposed to adapt the phase-sensitive electric ﬁeld

cross-correlation (EFXC) technique26 often referred to as

coherent optical sampling27 or dual-comb spectroscopy

(DCS)28 for imaging the dynamics of solitons in microres-

onators3,4. However, high scan rates with EFXC are

obtainable only with multi-GHz repetition-rate sources.

In the case of ﬁber laser cavities, which constitute a ma-

jority of SM generators, the aliasing (Nyquist) limitation

strongly constraints the observable optical bandwidth

due to sources’ low, MHz repetition rates, and restricts

frame rates to the 10’s–100’s of Hz range. The need

for phase locking between the lasers also adds a layer of

complexity. The greatest diﬃculty, however, results from

the need of a second, spectrally-matched laser, which may

be impractical to implement at exotic wavelengths.

To bypass these limitations and unlock the kHz- to

sub-MHz rate imaging potential, in this Article we adapt

the non-interferometric intensity cross-correlation (IXC)

technique29 to the problem of dynamic soliton imaging.

arXiv:2210.09966v1 [physics.optics] 18 Oct 2022

LPF

No locking needed

IXC –2PA

Optical Oscilloscope

Laser under

test (LUT)

Probe (local

oscillator,

LO)

fr+fr

-field

Time

0

2

−

1/(

r+

r)LO

LUT

EFXC (Conventional DCS):

IXC (2P DCS):

Intensity

Time

LUT

PD:

LNA LPF

Valence band

Conduction band

Virtual

state

PD material

bandgap

hνLUT

hνPROBE

2PA detection process

LO LUT

1550 nm 1150 –2200 nm

Si PD

Temporal

resolution

Frame

rate

Spectral

resolution

EFXC IXC DFT

kHz

MHz

GHz

MHz

Wavelength

flexibility

Low

High

IAC

Access to

molecular phase

μs

Expect.

Difficult

Figure 1 –Principles and experimental realization. a Conceptual schematic showing the laser under test (LUT) combined with a probe. After

the long-pass-ﬁltered (LPF) optical signal is measured on a photodetector (PD), a low-noise ampliﬁer (LNA) followed by a low-pass ﬁlter (LPF) are

used for signal conditioning prior to sampling by an oscilloscope. bComparison of sampling in the EFXC, and IXC. The lag increases linearly from

pulse-pair to pulse-air to produce EFXC and IXC signals. cEnergy diagram of the 2PA detection process. dChart-style comparison of existing SM

characterization techniques in conventional realizations with the IXC technique.

Two-photon dual-comb IXC (or simply IXC throughout

the rest of the text) probes SM dynamics with eased re-

strictions on the laser design, operation wavelength and

stability. Instead of nonlinear crystals30,31 that require

optical phase matching, the imaging technique builds on

the two-photon detection ranging concept by Wright et

al.32. Unlike EXFC, the IXC technique lifts the require-

ments of spectral overlap and phase lock between the two

lasers: a pair of free-running sources with diﬀerent wave-

lengths and oﬀset repetition rates can be used instead.

Only the probe and LUT photon energies must add to

satisfy the 2-photon-absorption (2PA) detection criterion,

which grants access to probing lasers in emerging spec-

tral regions. Such an implementation of IXC may also

extend the wavelength capabilities of techniques that re-

trieve the pulse temporal intensity proﬁle (not just IAC)

like FROG33 or SPIDER34, or speed of wavelength-agile

cross-correlation FROG (XFROG)35. If that of the probe

pulse in IXC is characterized and known, the tested pulse

proﬁle one can be retrieved via deconvolution29.

Results and discussion

Two-photon intensity cross-correlation. To image

the SM behavior in an exemplary laser-under-test (LUT)

cavity, we have probed it by a second mode-locked probe

laser (referred to as a local oscillator, LO). On average,

the lasers had tens of mW of optical powers and oﬀered

sub-ps pulse widths. The experimental setup is shown in

Fig. 1a. Both sources were combined with a ﬁber cou-

pler to jointly illuminate a commercial bias-free Si pho-

todiode (PD, Thorlabs FDS02), which exhibits a 2PA re-

sponse above ∼1100 nm. To prevent one-photon absorp-

tion from occurring (i. e. due to residual above-bandgap

pump or spurious light generated via nonlinear frequency

conversion), we have incorporated a ﬁber long-pass ﬁlter

(LPF) before the PD. The LUT repetition rate was fr,1≈

100.04 MHz, while the probe laser with fr,2=fr,1+ ∆fr

was equipped with a tunable delay line to vary the cavity

length and hence adjust the repetition rate diﬀerence ∆fr

governing the IXC scan rate. The basic principle of the

the IXC technique (Fig. 1b) resembles conventional DCS

(EFXC), where a time lag of ∆T≈∆fr/f2

rthat linearly

increases from pulse pair to pulse pair36 stroboscopically

samples the waveform over optical delays between 0 and

1/fr. In the case of dual-comb interferometry, measured

is the phase-sensitive E-ﬁeld cross-correlation also known

as the interferogram (IGM), which relates the eﬀective

(ps/fs), and laboratory time (µs/ns) time scales by the

temporal magniﬁcation factor (TMF) m=fr/∆fr. How-

ever, the Nyquist frequency located at fr/2 sets a limit on

the maximum probed optical bandwidth ∆νexpressed as

∆fr≤f2

r/2∆ν, above which the probed IGM is badly dis-

torted. Here, the full −20 dB bandwidth ∆ν= 7.5 THz

(60 nm) can be probed with EFXC at rates ∆fr≤670 Hz.

In IXC, the sampling process remains the same, however,

the interaction relies on multiplying the combs’ E-ﬁeld

intensities29 I1,2(t) = |E1,2(t)|2. The measured quantity

is simply the intensity cross-correlation (?): hI1(t)I2(t+

τ)i=R+∞

−∞ I1(t)I2(t+τ) dt=I1(t)? I2(t), where τrepre-

sents the lag scanned by the asynchronous interaction. As

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Two-photonimagingofsolitondynamicsukaszA.Sterczewski1;,andJarosawSotor11FacultyofElectronics,PhotonicsandMicrosystems,WroclawUniversityofScienceandTechnology,Wyb.Wyspianskiego27,50-370Wroclaw,Polande-mail:lukasz.sterczewski@pwr.edu.plOpticalsolitarywaves(solitons)thatinteractinanon-linearsystemc...

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Two-photon imaging of soliton dynamics Łukasz A. Sterczewski1 and Jarosław Sotor1 1Faculty of Electronics Photonics and Microsystems Wroclaw University of Science and Technology Wyb.

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