Avinash K Singh

Kalyan Santra

Xueyu Song

Jacob W Petrich

Emily A Smith

Spectral Narrowing Accompanies Enhanced Spatial Resolution in Saturated Coherent Anti-Stokes Raman Scattering (CARS): Comparisons of Experiment and Theory

We demonstrate theoretically and confirm experimentally the mechanism by which spectral narrowing accompanies enhanced spatial resolution in a saturated coherent anti-Stokes Raman scattering (CARS) signal that is demodulated at the third harmonic (3f) of the pump modulation frequency (f). Under these modulation conditions, theory predicts a narrowing of the full-width at half-maximum (FWHM) of the CARS spectrum by a factor of 2.0 with respect to that of the spectrum obtained by demodulation at the fundamental frequency. Theory also predicts an improvement of spatial resolution by a factor of 1.7. Experimentally, narrowing of the FWHM of the CARS spectrum of 1,4-bis((E)-2-methylstyryl) benzene (MSB) crystals by a factor of 2.5 is observed upon saturation. Further experimental confirmation is provided from investigating diamond particles, for which spectral narrowing was enhanced by a factor of 2.8 and spatial resolution was enhanced by a factor of 2. Details of the mechanism and execution of the saturated CARS experiment are elucidated and limits to its applicability are suggested, one of which is the conclusion that the saturation approach is not suitable for extraction of harmonics beyond 3f . In this work we have developed a more comprehensive understanding of the correlation between the observed experimental results and experimental factors than has been previously reported.

INTRODUCTION

Achieving vibrational subdiffraction imaging is an ongoing quest [START_REF] Prince | Going visible: high-resolution coherent Raman imaging of cells and tissues[END_REF][START_REF] Graefe | Far-field super-resolution vibrational spectroscopy[END_REF][START_REF] Upputuri | Super-resolution coherent anti-Stokes Raman scattering microscopy with photonic nanojets[END_REF][START_REF] Gasecka | Resolution and contrast enhancement in coherent anti-Stokes Raman-scattering microscopy[END_REF][START_REF] Yonemaru | Superspatial-and -spectral-resolution in vibrational imaging via saturated coherent anti-Stokes Raman scattering[END_REF][START_REF] Würthwein | Saturated Raman scattering for sub-diffractionlimited imaging[END_REF] whose interest has been motivated by the successful implementation of fluorescence techniques like stimulated emission depletion (STED), [START_REF] Hell | Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy[END_REF] structured-illumination microscopy, [START_REF] Gustafsson | Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution[END_REF] stochastic optical reconstruction microscopy (STORM) [START_REF] Rust | Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)[END_REF] and photoactivated localization microscopy (PALM). [START_REF] Betzig | Imaging intracellular fluorescent proteins at nanometer resolution[END_REF] These methods have made it possible to probe samples with features that are sub-10 nm on a routine basis. The fluorescence technique most relevant to our discussion exploits the saturated excitation of fluorescence (SAX), pioneered by Fujita and coworkers. [START_REF] Fujita | High-resolution confocal microscopy by saturated excitation of fluorescence[END_REF][START_REF] Nawa | Saturated excitation microscopy using differential excitation for efficient detection of nonlinear fluorescence signals[END_REF][START_REF] Yamanaka | SAX microscopy with fluorescent nanodiamond probes for high-resolution fluorescence imaging[END_REF][START_REF] Yonemaru | Saturated excitation microscopy with optimized excitation modulation[END_REF] SAX provides enhanced spatial resolution by using the nonlinearity induced in the fluorescence signal at higher incident laser powers. In principle, its spatial resolution is limited only by the highest harmonic that can be measured. In practice, the limitation on spatial resolution is imposed by the shot noise of the detector, which becomes increasingly significant as the signal strength decreases with successively higher harmonics. [START_REF] Fujita | High-resolution confocal microscopy by saturated excitation of fluorescence[END_REF] Although fluorescence techniques are successful in revealing intricate details, their application often requires the use of extrinsic organic fluorophores for labeling samples, especially in biological applications. This introduces two limitations. First, the experiment provides information on the system indirectly through the label. Second, labels can photobleach under laser irradiation and, more importantly, may be cytotoxic. [START_REF] Ettinger | Chapter 5 -Fluorescence live cell imaging[END_REF] Vibrational imaging provides a powerful alternative because it is able to exploit the chemical functionalities already present in the sample. This presents the added benefit of eliminating additional sample preparation.

The advent of coherent Raman techniques has contributed to the popularization of labelfree vibrational imaging. [START_REF] Zumbusch | Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering[END_REF][START_REF] Freudiger | Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy[END_REF] Coherent anti-Stokes Raman scattering (CARS) and stimulated Raman scattering (SRS) are the two most widely used coherent Raman spectroscopies, and they have been successfully implemented in various laboratories. [START_REF] Evans | Coherent anti-Stokes Raman scattering microscopy: chemical imaging for biology and medicine[END_REF][START_REF] Day | Quantitative coherent anti-Stokes Raman scattering (CARS) microscopy[END_REF][START_REF] Tipping | Stimulated Raman scattering microscopy: an emerging tool for drug discovery[END_REF][START_REF] Polli | Broadband coherent Raman scattering microscopy[END_REF][START_REF] Syed | Raman imaging in cell membranes, lipid-rich organelles and lipid bilayers[END_REF][START_REF] Rodriguez | Coherent anti-stokes Raman scattering microscopy: A biological review[END_REF] In addition to providing much larger signals than those of traditional spontaneous Raman spectroscopy, they are label-free techniques; and the near infrared wavelengths that are usually used for CARS and SRS penetrate deeper into tissues compared to the wavelengths used for fluorescence imaging. The penetration depth of the applied wavelengths is obviously a critical parameter for imaging thick biological samples. [START_REF] Sordillo | Deep optical imaging of tissue using the second and third near-infrared spectral windows[END_REF] Theoretical and experimental studies have already been performed in the area of subdiffraction coherent Raman microscopy. [START_REF] Yonemaru | Superspatial-and -spectral-resolution in vibrational imaging via saturated coherent anti-Stokes Raman scattering[END_REF][START_REF] Beeker | A route to sub-diffraction-limited CARS Microscopy[END_REF][START_REF] Cleff | Stimulated-emission pumping enabling sub-diffraction-limited spatial resolution in coherent anti-Stokes Raman scattering microscopy[END_REF][START_REF] Beeker | Spatially dependent Rabi oscillations: An approach to sub-diffraction-limited coherent anti-Stokes Raman-scattering microscopy[END_REF][START_REF] Silva | Toward label-free super-resolution microscopy[END_REF][START_REF] Kim | Superresolution four-wave mixing microscopy[END_REF] Silva et al. [START_REF] Silva | Toward label-free super-resolution microscopy[END_REF] demonstrated the elimination of the vibrational signal from the edges of a diffraction-limited spot using a doughnut-shaped decoherence pulse. This resulted in a focal spot smaller than the diffraction limit. Bi et al. employed visible wavelengths instead of near infrared wavelengths for SRS imaging by frequency doubling the pump and the Stokes beam; spatial resolution down to 130 nm in a near-resonance enhanced SRS microscopy experiment was reported. [START_REF] Bi | Nearresonance enhanced label-free stimulated Raman scattering microscopy with spatial resolution near 130 nm[END_REF] As an extension of SAX, Yonemaru et al. used saturation of CARS at high power densities to induce nonlinearity in the CARS signal leading to the generation of higher harmonics of the excitation pulse modulation frequency. [START_REF] Yonemaru | Superspatial-and -spectral-resolution in vibrational imaging via saturated coherent anti-Stokes Raman scattering[END_REF] A lock-in amplifier demodulated the higher harmonics of the signal frequency, and the image constructed by the third harmonic of the signal provided better spatial resolution than that constructed using the fundamental frequency. Enhanced spectral resolution was also observed.

Because of the high laser intensities required for these experiments, designed to improve spatial resolution, a common feature is to test them with what are traditionally believed to be robust, highly photostable samples, i.e., diamond particles [START_REF] Yonemaru | Superspatial-and -spectral-resolution in vibrational imaging via saturated coherent anti-Stokes Raman scattering[END_REF][START_REF] Silva | Toward label-free super-resolution microscopy[END_REF] . In this work, however, in order to examine the suitability of the saturated CARS technique to other chemical systems and obtain a thorough and rigorous physical understanding of the experimental observations, we have studied a model organic molecule, 1,4-bis(2-methylstyryl)benzene (MSB), in addition to diamond particles. (Because MSB was interrogated as a single crystal, it could only provide information on the spectral characteristics of the technique, while the diamond samples provided both spectral and spatial information.) Our results, limited to data collected at the third harmonic (3f) of the modulation frequency (f), show that both the spectral and spatial resolution is improved by a factor of ~2 with respect to that collected at the fundamental and second harmonics. Furthermore, our calculations indicate that enhancement cannot occur at the second harmonic. This is also confirmed experimentally. Finally, an important practical observation of this work is that it was necessarily limited to the third harmonic. This is because the MSB signal was too weak to observe at the fourth harmonic and because laser intensities required to observe signal from diamond at the fourth harmonic often destroyed the sample, despite its putative robustness.

EXPERIMENTAL SECTION

Sample Preparation

1,4-Bis(2-methylstyryl)benzene (MSB) (Figure 1) was purchased from Sigma-Aldrich and was recrystallized using toluene. A single crystal was mounted between two clean microscopy coverslips (Corning, thickness 0.16 -0.19 mm) and sealed with nail varnish. Diamond particles were purchased from Tomei Diamonds (IRM 2-4, Tokyo, Japan). A ~ 2 mg/mL solution of the diamond particles was drop-casted on a clean cover slip and air dried. The diamond particles employed have an average diameter of around 2-3 µm and were identical to those used by Yonemaru et al..

Instrumentation

Spontaneous Raman Spectroscopy. Spontaneous Raman measurements were performed with a XploRA Plus Raman confocal upright microscope equipped with a Synapse EMCCD camera (HORIBA Scientific, Edison, New Jersey). The sample was excited with 16 mW of a 785-nm laser source focused on the sample with a 100×, 0.9-numerical aperture (N.A.) objective. Raman spectra were collected using a grating with 1200-groove/mm and with a total acquisition time of 1 second for MSB and 3 seconds for diamond particles. An average of two accumulations is reported.

CARS Apparatus: Modulation of the Intensity of the Pump Beam. The schematic for our CARS microscopy apparatus is presented in Figure 2 and is based on the layout reported by Fujita and coworkers. [START_REF] Yonemaru | Superspatial-and -spectral-resolution in vibrational imaging via saturated coherent anti-Stokes Raman scattering[END_REF] It is based on a one-box picosecond laser source (PicoEmerald S, A.P.E Berlin) with a repetition rate of 80 MHz and a pulse width of 2.3 ps. It provides a tunable pump beam (720-990 nm) and a Stokes beam fixed at 1031 nm. The pump and Stokes beams exit the laser aperture collinearly and are separated by a dichroic shortpass filter (Edmund optics part#86-695, cut off 1000 nm). We subsequently used a Mach-Zehnder interferometer to modulate the pump beam. [START_REF] Yonemaru | Superspatial-and -spectral-resolution in vibrational imaging via saturated coherent anti-Stokes Raman scattering[END_REF] First, the pump beam was split using a 50:50 nonpolarizing beam splitter (Edmond optics part# 47-012). Each arm of the split pump beam was passed through an acousto-optical modulator (AOM) (AOM-402AF3, IntraAction Corp, Bellwood). One AOM was driven at 40 MHz and the other at 40.50 MHz using a dual-channel RF frequency driver (DFE-404A4, IntraAction Corp).

The first-order diffracted beams from both the AOMs were then recombined using a second beam splitter to give a beam intensity modulated at the difference frequency of the two AOMs (500 kHz). The selection of a fundamental frequency of 500 kHz was guided by our goal to detect the weak 3f signal above the laser noise. As the theoretical studies discussed below suggest, generating the 3f signal requires saturation of the CARS; and the 3f signal is weaker than those at f and 2f.

Accordingly, experimental parameters had to be adjusted to optimize the signal strength: the laser power density at the sample; the N.A. of the condenser lens used for the collection of the signal; the detector gain, the lock-in amplifier sensitivity, or both. The imposition of the modulation frequency on the pulse train was monitored with an oscilloscope (Agilent Infiniium 54830B DSO) (Figure S1, supporting information). In order to obtain efficient diffraction of the pulse train (and consequently, modulation depth) and to obtain the highest beam quality, the pump beam was directed into the AOM with a 500-mm focal length lens. Identical lenses were used to collimate the first-order diffracted beams. A retroreflector mounted on a computer-controlled delay stage (Physik Instrumente, M-410.CG) was used in one of the arms of the interferometer for compensating the temporal mismatch between the two beams at recombination. After the recombination at the second beam splitter, one part of the beam was detected by a photodiode (Thorlabs DET025A) and served as the reference frequency (f) for the lock-in amplifier (Model SR865A, Stanford Research Systems). A variable neutral-density filter was placed in front of the photodiode to prevent saturation of the photodiode. The second part of the beam was recombined with the Stokes beam using a dichroic, long-pass filter (Edmond optics part#87-047, 1000-nm cut off). A second retroreflector mounted on a computer-controlled delay stage was used in the path of the Stokes beam to ensure temporal overlap between the pump and Stokes beams in cases where the 20-picosecond internal delay of the laser was not enough to compensate for the path difference between the two paths. After recombination, the pump and the Stokes beams were spatially filtered and expanded (8×) with a homemade beam expander to match the back aperture of the objective lens (Nikon Plan Apo, 60×, 1.27-N.A. water immersion) and fed into the back port of an optical microscope (Nikon Eclipse Ti2). The CARS signal was collected in the forward mode using a 1.4-N.A. oil immersion condenser lens (Nikon HNA-oil) and detected by a Hamamatsu C10508-01 avalanche photodiode (APD). A stack of shortpass filters (ET850sp-2p, Chroma Technologies, 850-nm cut off) and one bandpass filter (FF01-769/41-25, Semrock for MSB; FF01-810/10-25, Semrock for diamond particles) were used to reject the pump and the Stokes beam. The fundamental and higher harmonics of the CARS signal were detected and demodulated using the lock-in amplifier. The SR865A lock-in amplifier is capable of extracting up to the 99th harmonic of the reference frequency if nf < 4MHz, where n is nth harmonic and f is the reference frequency.

The CARS signal at f and the higher harmonics (2f and 3f) were extracted using the lock-in amplifier. CARS and saturated CARS spectra were constructed by detuning the pump wavelength in small steps (typically 0.2-0.4 nm) to cover the full range of the Raman band under consideration with a sufficient number of points and demodulating the signal at f, 2f and 3f. Image construction was performed by scanning the sample over the pump and Stokes beams using an xyz-piezo stage (Nano-LP200, Mad city labs) and demodulating the signal at f and 3f at each pixel. Data and image acquisition were performed using GPScan.VI, [START_REF] Ferrand | VI: A general-purpose LabVIEW program for scanning imaging or any application requiring synchronous analog voltage generation and data acquisition[END_REF] a Labview-based program for image scanning with a National Instruments multifunction I/O device (PCIe-6353). Fiji-ImageJ package [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF] was used for image analysis. High numerical aperture condenser lenses were used for signal collection for their large light gathering capability; they also have the additional advantage of reducing background caused by processes such as cross-phase modulation. [START_REF] Freudiger | Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy[END_REF] In addition to the different power requirements for the f and the 3f, the measurement of a reproducible signal with good signal-to-noise ratio required adjustment of the detector parameters and the lock-in settings. Depending on the alignment and the incident power, the f and 2f signals were readily measured with a low detector gain (30-50) compared to the 3f signal, which needed adjustment of the detector gain to higher values (150).

The lock-in time constant was set at 300 µs with a dwell time of 1 ms for the signal measurement at f, 2f, and 3f. In this work, we have limited our discussion only up to the 3f signal. For MSB, higher harmonics (> 3f) could not be detected under the maximum available power density in the sample plane (12 MW/cm 2 ). While in the case of MSB, sample damage did not occur at these power densities, it did for the diamond particles. Thus, sample degradation of the diamond particles prevented acquisition of data at harmonics greater than 3f.

RESULTS AND DISCUSSION

Theory

The basis of the enhancement of the spectral and spatial resolution in coherent anti-Stokes Raman scattering microscopy is the saturation of the vibrational population difference at high laser powers. [START_REF] Yonemaru | Superspatial-and -spectral-resolution in vibrational imaging via saturated coherent anti-Stokes Raman scattering[END_REF] This intrinsically nonlinear effect [START_REF] Seigman | Lasers[END_REF] saturates the intensity of the CARS signal and induces additional nonlinearity in it. If, therefore, one of the lasers is modulated with a sinusoidal intensity variation, higher harmonics of that modulation frequency can be detected in the CARS signal. Here we present a detailed theoretical account of the saturation effect on the enhancement of the spectral and spatial resolution enhancement in CARS microscopy.

The intensity of the CARS signal ( ) CARS I generated at the focal plane (defined by 0 z = ) as a result of the interaction of two pump ( ) p photons and one Stokes ( ) s photon can be expressed as: [START_REF] Marrocco | Spectral synthesis of coherent anti-Stokes Raman spectra with broadband Stokes lasers of arbitrary line shapes[END_REF][START_REF] Zheltikov | Coherent anti-Stokes Raman scattering: from proof-of-the-principle experiments to femtosecond CARS and higher order wave-mixing generalizations[END_REF][START_REF] Yueh | Analytical expressions for coherent anti-stokes raman spectral (CARS) profiles[END_REF] (3)

2 2 ( , , ) | ( , , ) | [ ( , , )] ( , ) CARS a a pm p s s I t k t I t I ω χ ω ω ω = r r r r , (1)
where a ω , p ω and s ω are the frequencies of the anti-Stokes, pump, and Stokes beams, respectively. pm I and s I are the intensities of the pump and Stokes beams, respectively. [START_REF] Upputuri | Super-resolution coherent anti-Stokes Raman scattering microscopy with photonic nanojets[END_REF] χ is the third-order nonlinear susceptibility.

( , )

x y ≡ r at 0 z = are the spatial coordinates at the sample plane. t is the timescale of the intensity modulation of the pump beam. (Note that while the experiment described above relies upon pulsed laser beams, the only time dependence that we explicitly consider in our treatment is that arising from the modulation of pm I , which in turn modulates [START_REF] Upputuri | Super-resolution coherent anti-Stokes Raman scattering microscopy with photonic nanojets[END_REF] χ and CARS I

. Any intensity variations occurring on a much shorter timescale, such as those characterizing the profiles of the laser pulses or the oscillations of the optical fields, are integrated out over the modulation timescale t and can therefore can be ignored.)

2 0 (2 / ) k cn ε = is a proportionality constant (related
to the conversion of the electric field to intensity); c , the speed of light; 0 ε , the permittivity of light in a vacuum; and n , the refractive index of the medium. The spatial intensity distributions of the two beams on the xy -plane at the focal spot (

0) z = are: 2 2 ,0 2 2 1 ( , ) exp 2 2 j j j j x y I x y I πσ σ     + = -             , (2)
where j pm = or s for the modulated pump and Stokes beams, respectively. ,0 j I is the integrated intensity on the xy -plane at 0 z = . We assume circular symmetry of the laser beam at the focal spot with j σ as the beam-width parameter. Although our calculation is based on the focal plane, it can be generalized by incorporating the interaction length of the beams along the propagation direction (𝑧𝑧 axis) into equation (1). [START_REF] Tolles | A review of the theory and application of coherent anti-Stokes Raman spectroscopy (CARS)[END_REF][START_REF] Druet | Cars spectroscopy[END_REF] In our experiment, however, we did not vary the relative focus of the two beams in the z direction, and the collinear CARS signal from the entire focal volume was collected. This is equivalent to integrating the signal in the focal volume, which eliminates the zdependency in equation (2). Also, we note the laser beam is not monochromatic and thus has a finite spectral width. [START_REF] Marrocco | Spectral synthesis of coherent anti-Stokes Raman spectra with broadband Stokes lasers of arbitrary line shapes[END_REF] Since, however, we do not spectrally resolve the CARS signal but, rather, collect all the light generated at given central wavelengths of the laser beams, the effect is equivalent to that of integrating the signal in the spectral domain of the laser. We, therefore, omit the spectral width of the laser in the model.

The third-order nonlinear susceptibility is:

(3) (3) (3) ( , , ) ( , , ) a R a NR t t χ ω χ ω χ = + r r , (3)
where R and NR denote the resonant and nonresonant contributions. [START_REF] Moya | Gas concentration measurement by coherent Raman anti-Stokes scattering[END_REF][START_REF] Levenson | Dispersion of the nonlinear optical susceptibility tensor in centrosymmetric media[END_REF][START_REF] Shen | Theory of stimulated Brillouin and Raman scattering[END_REF][START_REF] Penzkofer | High intensity Raman interactions[END_REF] The resonant component [START_REF] Upputuri | Super-resolution coherent anti-Stokes Raman scattering microscopy with photonic nanojets[END_REF] ( , , ) a t χ ω r is:

4 (3) 0 4 2 ( , , ) ( , , ) 2[ ( )] a R a s v p s Nc t d t d i ε ω σ χ ω ω ω ω ω ∆   =   Ω - - -Γ    r r , ( 4
)
where N is the number density;

d d σ Ω
, the Raman cross section; Γ , the linewidth of the vibrational transition; v ω , the frequency of the vibrational transition; and  , the reduced Planck constant. ∆

is the normalized population difference between the two vibrational states involved in the transition (i.e., if there is no saturation 1 ∆ = ; it is 0 for complete saturation). The nonresonant contribution imparts asymmetry to the CARS spectrum, which is shown in Figure S2, supporting information. Combining equations (1), ( 3) and (4) yields:

2 2 2 (3) (3) 2 2 1 2 1 ( , , ) [( )( ( , , )) 2 ( , , ) ( ) ][ ( , , )] ( , ) CARS a a NR a NR pm p s s I t k a a t a t I t I ω ω χ ω χ ω ω = + ∆ + ∆ + r r r r r , (5)
where

( ) ( ) 4 0 1 2 4 2 2 2 4 v p s s v p s Nc d a d ω ω ω ε σ ω ω ω ω   - -     =   Ω     - - + Γ    , (6) and
( )

4 0 2 2 4 2 2 4 s v p s Nc d a d ε σ ω ω ω ω Γ   =   Ω     - - + Γ    . ( 7
)
The saturation of the population difference induces additional nonlinearity in the CARS signal via

(3) R χ . Because of stimulated Raman scattering (SRS), ∆ becomes strongly dependent on the intensity of the pump and the Stokes lasers. [START_REF] Yonemaru | Superspatial-and -spectral-resolution in vibrational imaging via saturated coherent anti-Stokes Raman scattering[END_REF][START_REF] Magnotti | Saturation and Stark broadening effects in dual-pump CARS of N2, O2, and H2[END_REF] The temporal dependence of ∆ on the modulation frequency can be described using the following kinetic equation:

( ) 2 2 2 2 ( , , ) 1 4 2 ( , , ) ( , ) ( , , ) 4
a pm p s s a s v p s d t c d I t I t dt d ω π σ ω ω ω ω ω ω ω   ∆ Γ   = -    ∆ Ω       - - + Γ    r r r r . (8)
Note that equation ( 8) only considers one of several processes that contribute to the population difference (other factors such as CARS and the lifetime of the excited vibrational state have been ignored owing to their smaller contribution). [START_REF] Druet | Cars spectroscopy[END_REF][START_REF] Magnotti | Saturation and Stark broadening effects in dual-pump CARS of N2, O2, and H2[END_REF] As described above, the intensity modulation of the pump beam is achieved via a Mach-Zehnder interferometer by mixing two frequency-shifted pump beams. The intensity of the modulated pump beam can be written as:

( ) , ( , , ) (1 cos ) 2 p p pm p m I I t t ω ω ω = + r r , (9)
where

2 m f ω π =
is the modulation frequency. The modulation of ∆ in equation ( 8) depends on the time-dependent part of pm I (i.e., we neglect the exponential decay of the population due to the constant term because the population of v = 1 is reasonably assumed to be depleted on a time scale that is much faster than the time scale of modulation) and can be written as:

( , , ) 1 cos ( , , ) a m a d t t t dt ω β ω ω ∆ = - ∆ r r , (10)
where ( )

2 2 2 2 ( , ) 4 2 ( , ) 2 4[ ] p p s s s v p s I c d I d ω π σ β ω ω ω ω ω   Γ   =     Ω - - + Γ      r r . ( 11
)
Solving the differential equation with the boundary condition 1 ∆ = at 0 t = , we have:

( , , ) exp sin a m m t t β ω ω ω   ∆ = -     r . ( 12
)
The effect on the population difference due to modulation of the pump beam is given in Figure 3 for various intensities of the pump beam. Substituting equations ( 9) and ( 12) in equation ( 5) yields: Mathematically, this can be expressed as:

( ) ( ) ( ) 2 2 2 (3) (3) 1 2 1 2 2
β β ω ω χ ω χ ω ω ω ω ω       = + - + - +                × +      
( , , ) [ ( , , )] CARS a CARS a I F I t ω ω = F r r , ( 14
)
where F denotes the modulation frequency space and F denotes the Fourier transform. Extraction (Because signal is collected at a given harmonic, the total signal at that harmonic is normalized by dividing all terms contributing to that signal at that harmonic by the magnitude of the largest term.)

The most important conclusion drawn from this analysis is that 𝛽𝛽 determines both the spectral and the spatial dependence of the signal. In other words, the maximum of the harmonic component, set to 1 and demodulated at the third harmonic, will show spectral as well as spatial narrowing compared to that corresponding to the normalized fundamental and second harmonic (which are identical to each other in these respects). This will lead to resolution enhancement in both the spectral and spatial domains when signal is collected at 3f or higher harmonics. Therefore, spatial resolution enhancement is fundamentally linked to the spectral resolution enhancement through the saturation of the population difference. These calculated results are presented in Figures 4 and5 for MSB and Figures S4 and S5 of SI for diamond particles. Our model implies limits on the laser power. If it is sufficiently great that higher powers of m β ω dominate the expansion, the terms corresponding to the third harmonic signal diminish in relative importance with respect to those of the fundamental and second harmonic; and spectral and spatial enhancement at the third harmonic become insignificant. Under our experimental conditions, m β ω is ~ 0.002. Thus, higherorder terms can be neglected; and we should, therefore, expect spectral and spatial enhancement at the third harmonic. Our experimental data confirm our theory and calculations, as shown in Figures 6 and7.

Enhancement of Spectral Resolution in Crystals of the Organic Molecule, MSB

The Raman cross section of MSB is estimated to be 65 times that of neat benzene. [START_REF] Mccreery | Raman spectroscopy for chemical analysis[END_REF] The 1593 cm - 1 band of MSB was selected for the CARS experiment as it is the most intense (Figure S6, supporting information). The CARS spectra were constructed by demodulating the signal at the fundamental (f, 500 kHz), the second harmonic (2f, 1 MHz), and the third harmonic (3f, 1.5 MHz).

The large MSB Raman cross section enabled the construction of the CARS spectrum by demodulating the signal at f and 2f using a pump power density of 0.8 MW/cm 2 and Stokes power density of 0.3 MW/cm 2 . The robustness of our theoretical analysis is further demonstrated in that it predicts that CARS signals at both f and 2f do not require saturation (Figure S3) and that the computed spectra (Figure 4) are identical, as is demonstrated experimentally (Figure 6). ; that for the 3f spectrum is 4.70 cm -1 . The parameters used for the calculation of the spectra and spatial profiles of saturated CARS whose results are presented here and in Figure 5 are the same as those used for Figure 3. Owing to the absence of any values for [START_REF] Upputuri | Super-resolution coherent anti-Stokes Raman scattering microscopy with photonic nanojets[END_REF] NR χ for MSB, we have used a value typical of organic molecules, 10 -14 e.s.u or 10 -22 m 2 /V 2 . [START_REF] Jr | Two-photon absorption measurements in organic liquids via nonlinear light mixing spectroscopy[END_REF] Furthermore, for the same reason, this value has been used in our computations of the signal for diamond (Figures S4 andS5, supporting information). The CARS spectrum is broader than the spontaneous Raman spectrum of MSB at power densities as low as 0.8 MW/cm 2 (Figure 6). Yonemaru et al. have also reported broadening of CARS spectrum for the 1332 cm -1 band of diamond constructed by demodulating the signal at the fundamental frequency. [START_REF] Yonemaru | Superspatial-and -spectral-resolution in vibrational imaging via saturated coherent anti-Stokes Raman scattering[END_REF] In their experiment, however, they maintained the same power levels for collecting f and 3f data and have attributed the broadening at f to saturation. In our case, for MSB we observed broadened CARS spectra at f when the pump power density was maintained either at a comparatively low excitation power density of 0.8 MW/cm 2 or a higher excitation power density of 2.8 MW/cm 2 , which was used to extract the saturated CARS signal at the third harmonic (Figure S7, supporting information). The broadening of the CARS spectrum at f can be attributed to the fast dissipation of the vibrational energy [START_REF] Fast | Coherent Raman scattering with plasmonic antennas[END_REF] in the MSB crystal as well as to the nonresonant background contribution, which is evident from the asymmetric CARS spectrum. ; CARS spectra at the fundamental (f, 500 kHz, blue) and the second harmonic (2f, 1MHz, olive); and the saturated CARS spectrum at the third harmonic (3f, 1.5 MHz, black). The pump and Stokes power densities for the CARS spectra at f and 2f were 0.8 MW/cm 2 and 0.3 MW/cm 2 , respectively; while for the saturated CARS spectrum at 3f, the pump and Stokes power densities were 2.8 MW/cm 2 and 1.6 MW/cm 2 , respectively.

For the detection of the 3f signal, the power densities used for the pump and the Stokes beams were 2.8 MW/cm 2 and 1.6 MW/cm 2 , respectively. In another set of experiments (Figure S8, supporting information), we were able to reproduce the trend using 2.3 MW/cm 2 and 1.3 MW/cm 2 of pump and Stokes power density, respectively, which indicates the onset of the 3f signal at somewhat lower power densities compared to those used for the 3f data presented in Figure 6 (black curve) .As shown in Figure 6, there is a significant narrowing of the 3f spectrum upon saturation. The FWHM of the f and 2f spectra are 18.3 cm -1 and 17.3 cm -1 , respectively. In comparison, the FWHM of the 3f spectrum is 7.2 cm -1 , which corresponds to a spectral narrowing of a factor of 2.5 compared to the theoretically determined factor of 2.0. Although the asymmetry and distortion of the f and 2f spectra induced by the nonresonant background make it difficult to assess precisely the extent of spectral narrowing, a comparison between FWHM of the 3f spectrum (7.2 cm -1 ) and the FWHM of the Lorentzian fit of the spontaneous Raman band (9.3 cm -1 ) are useful in making a comparison (Figure 6). Also, there appears to be suppression of the nonresonant background (NRB) in the 3f spectrum (as opposed to the f and 2f spectra), as has been previously demonstrated for diamond. [START_REF] Yonemaru | Superspatial-and -spectral-resolution in vibrational imaging via saturated coherent anti-Stokes Raman scattering[END_REF] While MSB provides a useful system for exploring the applicability of saturated CARS and for testing theoretical predications, it was only able to be investigated in the form of chemically homogeneous organic crystals, necessarily lacking discernible microstructures. Thus, it was not appropriate for interrogating the enhancement of spatial resolution. For this, we were obliged to resort to diamond particles.

Enhancement of Spectral and Spatial Resolution in Diamond Particles

Owing to their photostability, high number-density of tetrahedral carbon bonds, and biocompatibility, nonfluorescing diamond particles have shown promise for testing the development of new CARS imaging techniques. [START_REF] Yonemaru | Superspatial-and -spectral-resolution in vibrational imaging via saturated coherent anti-Stokes Raman scattering[END_REF][START_REF] Pope | Coherent anti-Stokes Raman scattering microscopy of single nanodiamonds[END_REF] Thus, we chose the 1332 cm -1 Raman band of diamond (Figure S9, supporting information) to examine the capability of our system for achieving enhanced spatial as well as spectral resolution. The diamond particles used had an average size of ~2-3 µm and were identical to those used by Yonemaru et al.. [START_REF] Yonemaru | Superspatial-and -spectral-resolution in vibrational imaging via saturated coherent anti-Stokes Raman scattering[END_REF] The spectra (Figure 7) and images (Figures 8a,b) corresponding to f and saturated 3f signals obtained by demodulating the signal at 500 kHz and at 1.5 MHz of the pump modulation frequency are presented. The data indicate spectral narrowing at 3f as compared to f. The FWHMs of the normalized spectra at f (14.8 cm -1 ) and 3f (5.2 cm -1 ), reveal a narrowing of a factor of 2.8. (We note that this determination was based upon the consideration of the central narrow feature of the 3f spectrum.) A visual comparison of the images obtained from the f and 3f data clearly shows improvements in spatial resolution (Figure 8c). As shown in Figure S4, the theoretically calculated spectral narrowing at 3f as compared to f is 2.0. Our experimental results are limited to the third harmonic. At the fourth harmonic, the signal from MSB was too weak to be detected; and the powers required to generate signal at the fourth harmonic often damaged the diamond particles. Thus, as has already been appreciated in the literature, there is a practical limitation to performing the saturated CARS experiment.

Recently, methods like differential SAX (dSAX) [START_REF] Nawa | Saturated excitation microscopy using differential excitation for efficient detection of nonlinear fluorescence signals[END_REF] and virtual sinusoidal modulation (VSM) [START_REF] Gong | Saturated stimulated-Raman-scattering microscopy for far-field superresolution vibrational imaging[END_REF] have been developed to improve the signal to noise ratio in saturation based techniques. dSAX, developed for saturated fluorescence microscopy has a better efficiency in extracting higher harmonic signal than regular SAX experiment for the same excitation power density thereby reducing the pixel dwell time and improving the signal to noise ratio. [START_REF] Nawa | Saturated excitation microscopy using differential excitation for efficient detection of nonlinear fluorescence signals[END_REF] Gong et al. have employed a similar strategy for high resolution stimulated Raman scattering imaging by VSM which alleviates the requirement of prolonged laser exposure to accumulate sufficient signal for image generation. [START_REF] Gong | Saturated stimulated-Raman-scattering microscopy for far-field superresolution vibrational imaging[END_REF] This scheme is particularly attractive as it helps prevent photodamage of the sample.

In addition, however, our analysis indicates that certain ratios of the laser power to the modulation frequency, quantified by the value defined as

𝛽𝛽 𝜔𝜔 𝑚𝑚

, can render the signal at the third harmonic negligible to those at higher harmonics. Given the high-power densities required to obtain a saturated CARS signal at the smallest possible harmonic, 3f, the utility of this technique is determined by maintaining a balance between the excitation power required for achieving the saturation while at the same time avoiding photodamage of the sample. In our experiments, in order to obtain the largest possible signals with which to test our theoretical results, we have employed strong Raman scatterers: MSB and diamond particles. But in the cases of other samples having functionalities with small Raman cross sections and low number densities for the bonds of interest, the incident power requirements may be prohibitively high. This will either require the reconsideration of experimental design or stimulate the development of algorithms for data analysis.

ASSOCIATED CONTENT

Supporting Information.

The Supporting Information is available free of charge on the ACS Publications website at DOI:xxxxxxxx.

Modulation waveform (Figure S1), Calculated CARS spectra for MSB (Figure S2), Variation of saturated CARS signal intensity with power for MSB (Figure S3), Saturated CARS spectra for diamond at f, 2f and 3f (Figure S4), Spatial distribution of saturated CARS at f, 2f and 3f for diamond (Figure S5), Spontaneous Raman spectrum of MSB (Figure S6), CARS spectrum at the fundamental frequency at different powers (Figure S7), Saturated CARS spectra of MSB at f and 3f (Figure S8), Spontaneous Raman spectrum of diamond particles (Figure S9).

Table of content (TOC) graphic

5

 5

Figure 1 .

 1 Figure 1. Structure of 1,4-bis((E)-2-methylstyryl) benzene.

Figure 2 .

 2 Figure 2. The CARS apparatus. M (mirror), DM (dichroic mirror), DL (delay line), L (lens), BS

Figure 3 .

 3 Figure 3. The effect on the population difference due to the modulation of the pump beam, at

  of a signal at a multiple of the modulation frequency can be accomplished via a lock-in amplifier using the modulation as the reference, as described above. The signals at the fundamental and its harmonics are obtained by collecting terms in the expansion corresponding to the appropriate powers ofm β ω. Specifically, the third harmonic will only contribute to the signal for nonzero powers of m β ω (as 3f originates solely because of the additional nonlinearity introduced by the saturation, always contributes to the signal corresponding to the fundamental and second harmonic and does not necessarily require saturation of the CARS signal.

Figure 4 .

 4 Figure 4. Saturated CARS spectra computed for MSB at f, 2f, and 3f show spectral narrowing

Figure 5 .

 5 Figure 5. Calculated spatial distribution of saturated CARS signal for MSB at f (a), 2f (b), 3f

Figure 6 .

 6 Figure 6. Comparison of experimentally obtained spectra for MSB: spontaneous Raman

Figure 7 .

 7 Figure 7. Spectra of diamond particles: spontaneous Raman spectrum (shaded pink region),

Figure 8 .

 8 Figure 8. (a) The CARS image of diamond particles collected at f. The pump and Stokes power

AUTHOR INFORMATION

Corresponding Authors

*E-mail : (E.A.S.) esmith1@iastate.edu *E-mail : (J.W.P.) jwp@iastate.edu *E-mail : (X.S.) xsong@iastate.edu

ORCID


Olivier Joly
email: olivier.joly@univ-lehavre.fr

Arnaud Le Marchand
email: arnaud.lemarchand@univ-lehavre.fr

Session 5: The Port Economy The Evolving Pattern of World Container Traffic Distribution

The evolving pattern of container traffic distribution is a highly stochastic phenomenon. Nevertheless, changes (hazard) in the distribution of port traffic do not seem to follow Gaussian laws. Shape of world container ports system, conceived as network geometry of container flows between ports explains both the interdependencies between ports performance and extreme fluctuations. Port traffic patterns are dependent results. The strength of interactions provokes a chaotic phenomenon within maritime container traffic network. External causes, even partial, need to be identified in order to evaluate local dynamics (for instance: shape of fluctuations for one given port by comparison with money fluctuations in the area (exchange rate volatility) or with the common trend in the range it belongs to). Otherwise, the continuity of container flows within the global maritime network could be more significant than singular 'port to port' interactions, given that network structures such as random, scale-free or hierarchical appear to be explaining factors.

Introduction: Various Factors

The determining factors of evolution and repartition of container traffics by ports and for each places, are multiple. Some factors are internal to each port system, and even from a local point of view, it is a complex phenomenon [START_REF] Goulielmos | Complexity theory applied to management of shipping companies[END_REF]. We cite productivity of handling operations, facilities for navigation, configuration of space, commercial politics and more widely the cohesion and the 'click effect' of the port community. The relations between these communities and the political decision-making centres 1 are summarized by the concept of governance of the port cities -matter in the mid-term and explain some factors of port performance that can be measured by containers flows. Seen by the outside, we can use the term of 'flexibility' of a community to analyse its internal efficiency, and the notion of 'reputation' to empathize the perception of this flexibility by the external actors. However, a harbour place is not apart. The regional and national economies have obviously effects on the port traffic throughputs. The trade balance, the growth of income and all current trends cannot be ignored to characterize the path of a given seaport. In the same way, the degree of opening, and the mode of insertion into the global economy plays a role of major influence. The strategy of major firms or even the strategy of relevant industrial districts are to be retained. If for instance, the firms were oriented towards exportation or delocalization, the consequences would not be the same for the port activity. In fact, we cannot consider the evolution of the ports independently of their context, which can be summarized into "situation and opening". However, between these key factors and the port efficiency, there are many retroaction loops 2 . Relations are not linear at this stage. Lastly, the global maritime container network, seen as a worldwide techno-economical system, is another sphere of determining factors for each seaport and for ports as a whole. The strategy of container shipping lines companies should not be taken into account at this scale. The mix of competition and cooperation between each seaport nodes of this global network is influencing each harbour places. Maritime traffic influences the activity of the port. In fact, the overall state of the global port network, especially its shape and its degree of connectivity are making waves on each port nodes…

The container port performances are a very complex subject and it is not amazing if it appears sometimes as a lattice of unpredictable black boxes. We need to improve our knowledge about relations between this lattice and the fluctuations of the port nodes.

Pathways and random walks

These remarks, already well known, lead us to think that in a statistical approach, container port traffic throughputs are considered as a stochastic phenomenon. It does not mean random change for each port. We can always find a reason, or a bunch of reasons, to explain why a given port has its activity growing fast, slow or decreasing. This can be analysed as result of the evolution of the professional relations, or public policies, by firm strategies or local trade. Nevertheless, it is still difficult to forecast because of the complexity of interactions that are engaged behind a rate growth of container flow. The pathways of the port systems and territories matters, but there are bifurcations, and the growth of a port follows usually a very broken line. In fact, we can say there is a random walk of the throughputs. We will show that the scale of traffic and the rank of one port among the others, are not explaining the rate growth. These variables are highly random like. That is compatible with the global distribution of the traffic that follows a Zipf-Pareto statistical distribution. This 1 as well as all the other problems 2 for instance between opening and governance of the territories phenomenon is explained by a network effect. Then, we will see that growth's fluctuations are non-Gaussian variables, with extreme values. This is due to a shape effect of the global maritime container traffic network. The spatial graph of global container's maritime traffic contributes to explain these characteristics [START_REF] Joly | Structuring of maritime traffic networks: Location of interconnected seaport platforms in North Western Europe[END_REF].

Scale -free growth rates

It is clear that the rank of ports does not determine their future rate growth. Figure 1 shows that the rank of the 184 biggest seaports in the world in 1990 never explains their growth during the decade up to 2001. The coefficient of determination R² is near zero (Figure 2) for the rate, and very low and not significant for the absolute performances. Note that in 2001 data, some major container ports 3 as Gioia Tauro for instance, were not present in the 1990 data 4 .

We can conclude from this simple estimate that rank and size are not discriminating factors. Same result is obtained for one year of variation. For instance, in 2001 some major ports have seen their traffics slightly decreasing, while smaller ports were growing. Such a result does not mean that size does not matter at all. It should be interpreted in relation with the Gibrat's statistical law as in industrial economics. The size of firms does not explain the growth rate. Therefore, we can assume that the evolution along the time is 'scalefree'. The advantages of size, (increasing returns, credit for new infrastructures), are in balance with the disadvantages (congestion, decreasing returns…). As written further, too many factors play a role in the competition, so the scale is only one key among others. This result is consistent with the attention paid to other studies on the institutions of the city-port communities. Because these institutions explain the port's flexibility and its ability to gain new container 3 'The port of Gioia Tauro is mainly a container terminal is nowadays one of the largest transhipment hub in the Mediterranean Sea (2003). 4 Gioia Tauro Container Terminal did not exist yet in 1990! flows. Furthermore, the major ports may continue to grow or not, depending on their efficiency and politics and not only from the past 5 .

Smaller ports can succeed to progress, depending on improvements and on a good context. The distribution of traffics is not definitely locked. What we qualify as ports' hazard the result of innovations in an open world. Innovations result from the complex interactions between flexibility, reputation and current trends.

The global distribution: the Zipf -Pareto statistical distribution

These previous observations are compatible with the distribution of traffic among ports, every year. This distribution is following, with a high degree of correlation, a well-known statistical law: the Pareto distribution. This law links the log of port rank with the log of size (measured by TEU Port Throughput). We can notice (Figure 4) that the coefficient of adjustment is quite good, the convexity into the left part of the line is usual and can be explained by some noise on the data for the weak performances. Such a distribution has some remarkable properties. It is possible to obtain such a shape by random simulations. It is independent of the scale of analysis: It is 'a fractal distribution' [START_REF] Mandelbrot | Fractales, hasard et finance[END_REF]Zajdenweber, 2000). One can note that if for the year 2001, we consider the 70 largest container seaports of the world, instead of the 280's; we find a correlation coefficient of 0.997. This may suggest the existence of 'a strong law among the core of the network'! 5 as resumed in the observed absolute or relative size of the traffics It is compatible with the possibility of very large extreme fluctuations. It suggests that we are in front of a chaotic and complex phenomenon [START_REF] Mandelbrot | Fractales, hasard et finance[END_REF]& Zajdenweber, 2000). It is a typical distribution among a large range of networks.

In addition, another technical point must be highlighted: the value of the parameter α6 in the estimation is minor to one, which means that7 there is no variance nor Esperance in such a distribution. There will be no convergence towards an average or a constant value for the variance among times. This leads us to the non-Gaussian fluctuations. It should be note that during the year 2001 and for the first 70 world top ranking container ports, the coefficient of determination R² equal 0.99.

Bubbles of containers and extreme fluctuations

If we look at the growth rate of the 50 biggest container ports during the year 2001, it is very easy to check that it does not follow a Gaussian law. The rates are too much scattered, and the extreme fluctuations are really far from the average. The first rate is 390 %8 the average is 10.2. Half the ports have negative rates. We have to represent the index of growth, rather than the rate. One remarkable point is that even the log-Gaussian law cannot figurate this dispersion. The tails are too heavy, the extreme are too frequents and it is not possible to put on the same table, Tanjung Pelepas, which is note on the right at the rank 60 when all the rest of the index are quoted less than 15. In fact, it is possible to find an adjustment: once again, the Pareto-Zipf law. If we take the index of variation for the 137 seaports, which have seen traffics, increase during 1990-2001 period. This property of the index's distribution means 'that we are in presence of extreme fluctuations, hyperbolics'. The parameters show that the variance of this index is infinite (Zajdenweber, 2000). So there is no convergence. Today it is similar to what we observe on the stock market exchange. One other remarkable point is that this result shows that the processes of localization in container traffics are not analysable in terms of the classical model of polarization, in which there is a stable equilibrium [START_REF] Arthur | Silicon Valley locational clusters: When do incrasing returns imply monopoly[END_REF]. This extreme dispersion of growth rate among the seaports has some correspondences. In the same way, by following the path of one port during a given time period, we can produce the same phenomenon. After years of growth, we can find a sudden stop or a negative rate. In fact, the path of any ports is always 'a broken line'. It seems that the evolution of some ports can be analysed as bubbles on stock exchange. They can depend on expectations and evolutions of reputation. If a harbour place 9 gives proof of improving organization (for instance), shipping companies might establish new lines on this port, making new links. That decision puts the related local traffic to increase, but if the recorded performance is not sufficient or slow after a while, that can lead to 'corrections', which provokes decrease, via complex agglomeration effects [START_REF] Scott | Les régions et l'économie mondiale[END_REF]. Overreactions appear. This 'bubbles' can be called 'mimetic or rational bubbles'. If it begins on an information problem 10 and spread like fashions, because of reputation effects, we can call it mimetic bubble. Such a phenomenon can be studied using the tools of the economics of convention [START_REF] Scott | Les régions et l'économie mondiale[END_REF]. Because of the reputation effects (which are following the construction of ports perception) which established some of the rules of prediction shared by actors. If it follows real changes, like innovations or political events, we can call it 'Rational Bubbles'. Case 11 where new information leads to correction, or bifurcation, lies between these notions. In a network approach, this kind of 'over-reaction' may be caused by the codependencies among the port places 12 . These co-dependencies can explain why the 'Gaussian law' is overflowing. The performances are not to be taken isolated, and they can be cumulative, retroactive, and they reflect chaotic evolutions of reputations and traffics.

It is a network's effect on its connectivity. The last decade was described by the concept of globalization. It is indeed 'a very hot context' for trade and maritime transport system, with important structural changes. The explosive upward trend in some Chinese ports are prominent examples, but they are not the only ones 13 . It is then conceivable that this context of fast evolutions, justifies the volatility of container flows and the pattern of distribution. It suggests some research should be focused on the links between this mobility and the volatility of other trends, like exchange rates. However, the shape of the spatial graph relative to global structuring of maritime container traffics gives confirmations. Extreme fluctuations, stochastic evolutions can be explained by some properties of the maritime container traffic network. It links distinct port ranges, highly clustered through short -cuts with regular and random vertices. It is a scale-free network, allowing a small world graph according to Watts' theory (Watt, 1998) and it is also a 'related 14 graph'. In such network, 'between order and randomness', data spread very quickly. Therefore, movements are faster than in fully random or completely hierarchical networks, the anticipations are "kaleidoscopic" and in consequence, the local pathways are partly unlocked. Thus, opportunities and risks still exist for each port community. It explains why some maritime shipping companies reach uncommon hubs (Figure 8).

Seaports, flows, merchant vessels & global maritime container traffics

Let us remind of a few formulations that could provide a common basis for the understanding of the maritime transport system and allow comparisons between systems in different regions of the world.

Behind the term of global maritime container traffic network 15 , we understand both a set of geographic locations namely container ports 16 that are interconnected -in the global container seaport system -by numerous segments of maritime routes and last but not least, by a set of 'interoceanic and transoceanic container flows' carried on by containerships. This suggests six main building blocks or major components:

(1) Ports of Origin, (2) Segments of maritime routes, (3) Respective channelled container flows, (4) Relay-ports, (5) Hub ports 17 and (6) Ports of Destinationlast call.

Thus, the name of 'maritime container traffic network' can be assigned to the framework of maritime routes within the world container seaport system and be used to describe tangible but invisible sea corridors. A maritime route is not simply a single shipping link between two interchange or relay ports; a maritime route is a part of a larger maritime network of sea corridors, is made up of several maritime segments made up of single shipping links between 2 (or more) calling ports as well.

15 or worldwide maritime container traffic network 16 or container terminals 17 That include the Sea-to-Sea Transhipment ports

(a) (b) (c)
Seaports or port nodes are points on the maritime network where both links, segments and routes converge, and often act as the focus for transport services and/or for the transhipment between two distinct modes of transport18 . Lastly, channelled container flows were worn along maritime shipping links by the containerships. Aggregations of continuous container flows along links between two ports constitute inter and transoceanic container traffic flows.

Using theoretical concepts [START_REF] Kansky | Structure of transportation networks, Relationship between network geometry and regional characteristics[END_REF] and network analysis tools that are often applied to inland goods transport networks, worldwide structuring of maritime container traffics has been studied using an original reliable source: Lloyd's Voyage Records data. Lloyd's Voyage Records is a weekly publication that enables to identify precisely worldwide from-port-to-port movements of numerous ocean-going merchant vessels19 including conventional and partmodified conventional vessels carrying deck loads of containers and purpose-built cellular container vessels. So we can locate where a container vessel was at a given time but also where it came fromits last port of call and where it was going toits next stop, and all this over a given period.

Almost 2000 successive sequences of port calls have been recorded20 .

The shape of the global container ports & shipping system

Graphs are a way of representing transportation networks linking ports together. According to graph-theoretic terms, the graph G is a set that contains two subsets of elements: (1) subset of vertices, and (2) subset of edges. An edge is an element of the graph G such as a continuous line between two distinct vertices. A vertex called a node or a point is an element of the graph such as a point of intersection of n edges.

Network geometry has become an important element in determining efficiency levels of container operations [START_REF] Slack | Shipping lines as agents of change in the port industry[END_REF]. One can see graph theory such as a starting point towards expected structural theory of network analysis centred on a set of geographical 'intertwined' locations [START_REF] Capineri | Italian geography and networks: a bibliographical essay[END_REF]. So we can consider 3 well-known network structures -useful in research into network topology and dynamics that are necessary to understand the evolution and behaviour of complex network systems [START_REF] Stocker | Network Structures and Agreement in Social Network Simulations[END_REF] chosen from many other classical network structures such as Global, Regional Hub-and-Spoke or even Centralized networks (Figure 8) :

-Hierarchical network structure represents classical spatial patterns: equilibrium spatial systems based on the complementarity relationships between market areas and nested ranked centres. They consist of nodes with branching connections that form a typical tree-like structure. According to graph-theoretic terms, a tree is a connected graph21 of at least two vertices such that graph does not contain any circuit (i.e.: finite path in which the initial vertex coincides with the terminal vertex of the path).

-Scale-free network consist of a few highly connected nodes that link the remaining sparsely connected nodes to the whole system [START_REF] Stocker | Network Structures and Agreement in Social Network Simulations[END_REF]. In terms of the classical spatial patterns, we can consider that scale-free networks are closed to multicentre networks (dynamic spatial systems, consisting of nodes with different functions that interact according to the circular cumulative mechanisms ruling the agglomeration and polarisation processes) [START_REF] Rabino | Understanding spatial structure from network data: Theoretical considerations and applications[END_REF].

-Simulated random networks have no apparent structure when nodes are placed randomly; the distribution of the number of each node connections approximates a Poisson probabilistic distribution.

In order to make possible the qualification and measuring of the global maritime container traffic network's structure and attributes, a few topological indices22 will be presented later.

Characterizing network geometry of container flows between ports: Findings

Anyway, Kansky wrote that the term of structure denotes the layout, geometry, or network pattern of transportation facilities or systems [START_REF] Kansky | Structure of transportation networks, Relationship between network geometry and regional characteristics[END_REF]. If these expressions imply a set of spatial relations between special components of global maritime container traffic network in respect with each other and to 'the whole organized', by measuring such relations, and according to Kansky, we should describe the structure of the global maritime container traffic network in technical terms. First, we present in the following two major data results relative to connected and strongly connected structure of global maritime container traffic network.

The global maritime container traffic network is connected 23

The graph relative to world maritime container traffic network is a finite connected graph.

During 1990 & 1992 autumn periods, it contained no significant isolated subgraphs24 and finite number of elements. There are no significant disconnected maritime networks within the global 888 container ports network during the 1990 survey period -respectively within the global 915 container ports network during the 1992 survey period (Figure 9). Note that during the 1990 survey period, except one major connected component grouping 880 container ports together, exactly 5 isolated subgraphs (3 made up of 2 isolated related ports component & 2 artefacts) have been identified and result from the original reliable Lloyd's Voyage Records source intrinsic data structure. They are not significant matter. Lastly, no artefacts are observed during 1992 survey period: all 915 observed ports belong to the only one connected component. Never container ports 25 are isolated from any global maritime traffic exchange. We assume feasible maritime paths exist between any ports of origin and destination in the global maritime network.

. Global maritime container traffic network is 'strongly connected'

'A connected graph' is said to be 'strongly connected' if for each pair of nodes i and j, there is a path starting at i and ending at j knowing that a path in 'a directed graph' is a sequence of some k nodes 26 with k ≥ 2. 'The connected graph' relative to the world maritime container traffic network is almost strongly connected (Figure 10). The results of the analysis indicate that according to 1990 survey period -within the major subset of 788 container ports 27 -any port or container terminal can be reached from any other port by following a path 28 consisting of a finite number of single maritime links.

According to 1992 survey period, number of ports increased up to 915 and ports belonging to the major strongly connected component also increased up to 839 [START_REF] Joly | Structuring of maritime traffic networks: Location of interconnected seaport platforms in North Western Europe[END_REF].

Other strongly connected components relative to 1990 period: 88 strongly connected components with only one port and six with a pair of ports are not significant at all: Lloyd's Voyage Records intrinsic structure 29 distorts the results. Each of t these (88 + 6) minor components should not be considered as a 'terminal black hole or end-of the-world' in the one hand, or spontaneous traffic generating origin port, in the other hand... Same remarks are relevant for the 1992 survey period. The structuring of global maritime container traffic network has one major strongly connected component of calling port call nodes within channelled container flows are blown along any maritime port-to-port links by the containerships, under constraints such as Estimated Time of Arrival 30 , nautical access and above all, the commercial strategies of shipping companies 31 . Knowing that we have to apprehend the degree of connectivity of such global maritime container traffic network, the notion of 'connected graph' provides us result about the continuity of relations between vertices of the graph representing the network. However, be careful, this notion of 'connected graph' lumps together their nature: a link between two vertices is equal to any path that make possible to join these two points with several transits. With the help of cycle and circuit notions 32 , graph-theoretic concept of connectivity specifies the nature of relationships and links between points. In order to quantify connectivity we present further Indices alpha α, beta β and gamma γ knowing that some scholars have observed the limits of graph theory topological measurements 33 are often too weak to differentiate among transportation networks.

26 node1, node 2, ... , node k 27 namely major stongly connected component or major nonconnected subgraph relative to major isolated network 28 That is to say maritime routes, segments or combination of continuous direct links 29 of calling port period sequences with missing logical feed-back 30 ETA 31 In other words: the shipping lines commercial ports strategies 32 A circuit is a finite path within a 'directed graph', in which the initial vertex coincides with the terminal vertex of the path. 33 and the difficulty to manage network with large number of nodes and edges

Connectivity measures of global maritime container traffic network

Indices alpha α, beta β and gamma γ are classical ratio measures of ''Whole Transportation Network'' expressing relations between distinguishable elements i.e. observed number of fundamental circuits / number of maximal circuits and edges / vertices [START_REF] Kansky | Structure of transportation networks, Relationship between network geometry and regional characteristics[END_REF]. The K cyclomatic number 34 is an arithmetic comparison between the number of port nodes, the number of maritime links between 2 calling ports and the number of strongly connected components 35 .

In the 1992 survey period, completely interconnected network 36 would accept a KMAX equal to 417.241. One can use K to determinate index α as an indicator of trellis or lattice and the geometry of the network as an adjusted form of K ratio 37 . Index α 38 can be interpreted as a percent of maximum connectivity of World maritime container traffic network. Values of index α are closed to 0: 0.0096% and 0.0101% (Figure 11) for each survey period according to K values: Connectivity of global maritime container traffic connected network is very weak. A few container ports related by connected network are belonging to circuits. 'Global maritime container traffic network trellis' 39 is loose. So assessment of the number of alternative paths within the world maritime traffic 40 blown along shipping links by the containerships demonstrates that the whole structuring of world maritime container traffic network is very weakly jointly liable by circuits or alternative circular paths. 7 Index  is the simplest form of connectivity measure, which is written as ratio between number of direct links, and number of port nodes, networks with complicated structure will have high values of, whereas networks with a simple structure will have low values [START_REF] Kansky | Structure of transportation networks, Relationship between network geometry and regional characteristics[END_REF].  could appear to be weak to differentiate among transportation networks. Calling Port Nodes 888 915 34 In a connected graph, the K cyclomatic number is equal to the maximum number of fundamental circuits. 35 i. e.: non-connected subgraphs 36 with complete connected graph with equal number of direct port-to-port links and port nodes 37 between observed number of circuits and maximum theoretical number of circuits 38 when the value is multiplied by 100 39 Or lattice 40 Namely maritime circulation of container merchant vessels Therefore, index  equals 5.24 during 1990 period and 5.64 during 1992 period, its whole scale is from zero to ∞ (for non-planar graphs such as air or maritime transportation graphs). It indicates that in the global maritime network, each container port is connected without transit to five ports. It is a kind of average degree of nodal incidence for the network that equals 5. This index  can also be partially improved by the 'observed directed 41 from-portto-port containerships movements' data' instead of 'directed from-port-to-port connections' without transit (Arcs). It indicates that during 50 dayslong 1990 autumn period and 50 days long 1992 autumn period, calling ports are reached by respectively almost 14 and 16 containerships! Connectivity of the global maritime container traffic related network is still very weak. However, index  results do not seem to be efficient and Gamma index  provides furthermore relevant results. Index gamma γ is a quotient of the observed number of edges to the maximum number of edges that lower limit is zero and upper limit is one assigned to all completely connected networks whether they have 5 or 5.000 vertices [START_REF] Kansky | Structure of transportation networks, Relationship between network geometry and regional characteristics[END_REF] 42 .

Our results for such indicator of trellis geometry of network: γ equal 0.0118 and 0.0123 43 for the two survey periods 1990 and 1992! Thus, γ is almost invariant between the 2 periods; it quantifies direct links within the whole network. So it indicates, by comparison with 'a theoretical ideal and fully interconnected network' 44 , that only a few calling ports are related without transit: 1.18% out of 888 ports and 1.23% out of 915 ports for the 2 survey periods 1990 and 1992! This translates into the fact that to link all the container ports of the world within an existing connected network, the 'direct interport relationships' 45 are not predominant: some transhipment ports relationships constitute a valid observation.

To conclude, our connectivity measurement underlines the structuring of global maritime container traffic network that is not jointly liable nor by maritime directed connections, nor circuits and alternative circular paths. In fact, a multipolar worldwide core of highly interconnected container ports wherein transhipment and traffic distributive operations are intensive appears linked to most of the remaining sparsely connected container ports.

9. An 'Invariant Core' for the Global Maritime Network (1990&1992 periods)

The configuration of the global maritime container traffic network seems to accept a core of highly interconnected hub container ports linking the remaining sparsely connected service container port nodes to the entire maritime traffic system. According to O'Kelly and Miller's review and synthesis, a hub network consists of three major components occupying a unique geographic location and these essential elements of singular hub transportation networks are located on the earth's surface in geometric patterns:

-'service nodes' such as points of specific location from which flows can originate and into which only flows which are destined for that specific location can enter; 41 i.e.: oriented vessels movements data 42 Note that Kansky named ratio 1/ γ: 'Degree of Connectivity'. 43 so close to 0 44 That is to say with complete connected graph bearing equal number of direct port-to-port links and port nodes 45 i.e.: port-to-port links without transit -'hub nodes' that have the characteristics of services nodes (i.e.: they can be flows origin and destination) but also allow the passage of through-flows or transshipment flows that are not intended to that location; -Lastly, 'arcs' 46 that connect the service nodes and hub nodes must have the two following properties:

(1) Every 'service node port' must be connected to, at least, one hub port node;

(2) A valid path must exist between all 'hub port nodes'.

These properties ensure that feasible path will exist between all origins and destinations in the network [START_REF] O'kelly | The hub network design problem -A review and synthesis[END_REF]. Incidence degree (of a given vertex) is a graph theoretical measure of individual elements of transportation networks, which provides the number of edges from a given vertex to each of the other vertices. Port incidence degree d(Port i) provides total number of from-port-to-port links from a given port to each of the others (Figure 13).

It shows that:

-Relayports 47 as d(Port i)  2 (including 'service node ports') represent a ports subset of 70 to 72% from total number of ports which do not belong to the core;

-Other Port incidence degree results provide 3classes of container ports which d(Port i) belong to ]10, 50], ]50, 100] & >100, constitute major part of 'the invariant container ports core' and represent minus 30% from total number of ports.

Within this minor subset, one can find major hub and loadcentre ports 48 that reaches sufficient volumes to sustain high efficiency shipping in a high density trade corridors with high connectivity level 49 , the former ports might be regarded as 'Mega terminals' [START_REF] Robinson | Asian hub/feeders nets: the dynamics of restructuring[END_REF] The main ports of the world, considered as a whole, admit a hierarchical spatial structure. Ranges or Regions hierarchy are then considered as a structure in their own right whose different parts are linked by predominance relationships [START_REF] Medda | Hierarchies of spatial networks systems. New Contributions to Transportation Analysis in Europe[END_REF]. Using 'the dominant-flow approach' as pioneered in 1961 by Nystuen and Dacey that involves a prior rule to define the 'centrality' of a place within a system [START_REF] Rabino | Understanding spatial structure from network data: Theoretical considerations and applications[END_REF], it makes possible to underline bipolarisation and organization into a hierarchy of 8 main world maritime traffic zones from A to H 53 ) in both terms of:

-the import and export dominant maritime areas and port ranges, -In addition, the satellite 54 maritime areas (Figure 14) and port ranges [START_REF] Joly | Structuring of maritime traffic networks: Location of interconnected seaport platforms in North Western Europe[END_REF] 55 .

However, within a given seaport range, there are many nonpredominant ports which are directly linked without any size determinism to various nonclosely related port ranges (Figure 14), even directly connected to single overseas container ports within a non-adjoining port range and such configuration define scale-free network structure (Watts, 1999[START_REF] Stocker | Network Structures and Agreement in Social Network Simulations[END_REF]. First part of this assumption is close to the 5th phase of Hayuth's 5 phases model for containerized port systems relative to: 'New direct service lines to some peripheral ports emerge, while the ocean trade route network is still composed of a relatively few consolidated long-distance routes' [START_REF] Wang | A container load center with a developing hinterland: A case study of Honk Kong[END_REF]. It follows that this configuration could well be useful to describe the early steps of scale-free network structure.

Concluding remarks

The main results can be summarized as follows:

(a) There is some degree of correlation between scale-free network structure of global maritime container traffic network and evolving pattern of world container traffic distribution. So because of the scale-free structure of this network and its strongly connected whole shape 56 , data and traffic flows spread quicker and easier than within classical hierarchical network or within weakly connected networks.

(b) This degree of correlation between the scale-free network structure of global maritime container traffic network and the evolving pattern of world container traffic distribution could be an explanation for non -Gaussian fluctuations & an explosive growth of container maritime traffic throughputs.

53 With reference to the practical and gobal spatial division of maritime areas proposed by the Reed's Marine Distances Table Editions. 54 or 2 nd rank maritime areas 55 See [START_REF] Joly | Structuring of maritime traffic networks: Location of interconnected seaport platforms in North Western Europe[END_REF] The Structuring of maritime traffic networks: Location of interconnected seaport platforms in North Western Europe. 56 Namely integrating the 'invariant core'of highly interconnected hub container ports linking the remaining sparsely connected service container port nodes to the whole maritime traffic system

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  Figure 2. 1990-2001 Growth rate

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  Figure 7. Rank -Size Law (Pareto) for positive Growth rates y = -1,4177x + 2,3004 R 2 = 0,973

Figure 8 .

 8 Figure 8. Network structures: (a) random, (b) scale-free and, (c) hierarchical (Stocker et al. 2001)

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  Figure 9: Related world maritime container traffic network

Figure

  Figure 10: Strongly connected world maritime container traffic network

Figure

  Figure 11. Graph-Theoretic Measures of Whole Transportation Network: Global maritime container traffic network

Figure 12 .From

 12 Figure 12. From-port-to-port connections & container ships movements' empirical data

Figure 13 :

 13 Figure 13: World Container Ports distribution according toThe i port's incidence degree: d(Port i)

Figure

  Figure 14 : 8 Main world maritime container traffic zones & Shipping routes: Scale-free network structure

13 bis: World Maritime Container flows in 1992 and Underlying Structure (Joly, 1999) 52

  -except the 2 world main nautical transit points: Panama Canal & the Straits of Gibraltar.It is clearly outstanding that 'the spatial significant 50 re-distribution of channelled container flows' is operated by a few container terminals and ports but with a wild range of local characteristics. Partial analysis of 2001 results seems to present a less than 200 container ports core which keep less than 25% (North Western Europa: A), less than 17% (Mediterranean: B), less than 16% (USEC, Caribbean & Gulf Coast: C) & more than 14% (Far East Asia: F) geographical distribution 51 .

	Figure

46

That is to say directed links without transit

47

As common seaports 48 1+3=4% of the overall ports 49 d(Port i) belonging to ]50, 100] & d(Port i) >100 50 Considered on a global scale 51 Sea below in the added Figure 13bis by O. Joly 2021 showing the representation of the geographical division of maritime areas according to the 1988 Reed's Marine Distances Table.

52 according to César Ducruet. La spatialité des réseaux maritimes : Contributions maritimes à l'analyse des réseaux en géographie. Géographie. Université de Paris 1 Panthéon-Sorbonne, 2016.

The shape parameter α determines the slope of the log-linear relationship between the (logarithms of) rank and size.

a very important characteristic

According to the Tanjung Pelepas Container Terminal

i.e.: connections with inland (hinterland) transportation networks

Vessels that appear in the records of the Lloyd's

relative to 1990 & 1992 Autumn time periods

i.e.: a graph that contains not isoled subgraphs associated with isolated subnetworks

classical graph theory indices

i.e.: a subgraph S is graph wich is contained in the graph G such that very element of the subgraph S is an element of G and some elements of the graph G are elements of the subgraph S