2.1.

Soliton Dynamics

The propagation of light in a PCF along its main axis $z$ is governed by the nonlinear Schrödinger equation (NLSE)²⁵

An analytical solution to the NLSE can be found in the case of anomalous dispersion (${\beta }_2<0$). This solution describes a pulse of light that propagates without changing its shape, known as a fundamental soliton. It can be expressed as follows:

For short pulses ($<5\mathrm{ps}$), the simplest form of the NLSE [Eq. (1)] is not sufficient to fully describe the behavior of solitons. Higher-order effects have to be taken into account, particularly intrapulse Raman scattering, which is responsible for the SSFS that we will use as a tuning mechanism. When the pulse is very short (typically $\approx 100\mathrm{fs}$), its spectrum is broad enough that an effect similar to stimulated Raman scattering can occur between the blue and the red components, mediated by the vibration modes of silica. Namely, the red part of the spectrum experiences a Raman gain, while the blue part experiences a Raman loss. This results in the continuous redshift of the soliton along its propagation in the fiber. The variation of the central frequency of the soliton is related to the duration of the pulse

When more power is injected into the fiber, the duration of the pulse decreases [see Eq. (3)], therefore, the redshift is enhanced. We propose to use this effect as a wavelength tuning mechanism. Indeed, since the redshift is proportional to $T_0^{-4}$, a small variation in the pulse duration will cause a large redshift and we can obtain a large tuning range for a virtually constant $T_0$.

If the third-order dispersion is small enough, the shape of the pulse remains the same during propagation and it is Fourier-transform-limited at any $z$ in the fiber, which means that it is the shortest pulse allowed by its bandwidth, with a constant time-bandwidth product $\mathrm{\Delta }{t}_{\mathrm{FWHM}}\mathrm{\Delta }{\nu }_{\mathrm{FWHM}}=4{[\ln (1+\sqrt{2})]}^2/{\pi }^2\approx 0.315$ characteristic for ${\mathrm{sech}}^2$ pulses. This is an interesting property for nonlinear microscopy as the contrasts we wish to activate require a high peak power, and therefore a short pulse duration.

2.2.

SC-PBG Fiber

From Eq. (3), we get that the energy of the soliton depends on the dispersion to nonlinear coefficient ratio ($|{\beta }_2|/\gamma$) which are the parameters of the fiber. We can rewrite it in the following way:

In order to get high excitation powers compatible with nonlinear microscopy, one has to fabricate a fiber whose design provides a high $|{\beta }_2|/\gamma$ ratio. We designed and fabricated an SC-PBG fiber with a doubly periodic cladding consisting of high index inclusions made of germanium-doped silica that are responsible for photonic bandgap guidance and air holes for reducing the confinement losses. Figure 1(a) shows a scanning electron microscope image of the fiber. This fiber resembled the one in Ref. 26, although the transmission window was optimized for 800 nm lasers. In the present SC-PBG fiber, the generation and redshift of solitons are possible in the first PBG, from the zero dispersion wavelength at 750 nm to the long-wavelength edge at 1030 nm. Such a fiber was shown to generate 3 to 5 times higher-energy solitons than standard PCFs.²⁷ Figure 1(b) shows the $|{\beta }_2|/\gamma$ ratio for the fiber we used in comparison to standard values for other PCFs. Coupling efficiencies were about 30% to 40% and the soliton generation efficiency, defined as the ratio of the soliton power over the total input power, was about 10%.

Fig. 1

(a) Scanning electron microscope image of the solid-core photonic bandgap (SC-PBG) fiber. Dark gray: silica; light gray: Ge-doped silica; black: air. (b) Red line: $|{\beta }_2|/\gamma$ ratios calculated for the SC-PBG fiber used in this work (finite element method simulation). Gray area: typical values of this ratio for standard PCFs.

2.3.

Optical Setups

2.3.2.

Spectroscopy and microscopy

We conducted two sets of experiments based on the excitation of the sample by the optical solitons generated in the fiber. First, we performed microspectroscopy on liquid samples, then we did microscopy on several samples, including biological ones.

The solitons were generated by a coupling 800 nm light from either an 80 MHz (Chameleon, 150 fs from Coherent Inc., Santa Clara, California) or a 1 GHz laser (GigaJet, 30 fs from Laser Quantum, Stockport, United Kingdom) into the SC-PBG fiber. As before, the soliton wavelength was controlled by changing the input power, and was filtered from residual and dispersive waves with a long-pass filter to ensure that the soliton was the only excitation pulse reaching the sample. For microspectroscopy, the light was focused on the sample with a long working distance objective ($20\times$, $\mathrm{NA}=0.45$ from Olympus, Tokyo, Japan) and collected by another objective ($\mathrm{NA}=0.60$) in the forward direction. We detected the TPEF on an APD after a set of long-pass and bandpass filters.

For microscopy, we used a high NA objective ($40\times$, $\mathrm{NA}=1.15$, water immersion from Nikon, Tokyo, Japan) and scanned the focus across the sample with galvanometric mirrors to create TPEF and SHG images in an epi-detection scheme by using two PMTs and an appropriate set of filters and dichroic mirrors.

A scheme representing the two setups combined is shown in Fig. 2.

Fig. 2

Scheme of the microspectroscopy and microscopy setup. PBS: polarizing beam splitter; $\lambda /2$: half-wave plate; DM: dichroic mirror; LP/BP filter: long-pass/bandpass filter; PMT: photomultiplier tube; and APD: avalanche photodiode.

3.1.

Generation of Solitons in the SC-PBG Fiber

An example of output spectrum is shown in Fig. 3(a). The red-shifted soliton is perfectly fitted by a squared hyperbolic-secant function (red dotted line). The inset of Fig. 3(a) shows the autocorrelation trace for the 900 nm soliton corresponding to the spectrum. For ${\mathrm{sech}}^2$ pulses, we divide the width FWHM of the autocorrelation trace by 1.54 in order to recover the pulse duration, which in this case was found to be 90 fs. The Fourier-transform-limited duration calculated from the Fourier transform of the spectrum ($\mathrm{\Delta }\lambda =10.7\mathrm{nm}$) is 80 fs. The difference can be explained by the dispersion of the optics between the fiber and the autocorrelator (lens, filter, halfwave plate), estimated to be about $1200{\mathrm{fs}}^2$. Thus we verify that the soliton is Fourier-transform-limited at the output of the fiber.

Fig. 3

(a) Black line: spectrum of the output of the fiber. Red dotted line: squared hyperbolic-secant fit of the soliton. Inset: autocorrelation trace of the 900 nm soliton. (b) Wavelength dependence of the energy of the soliton for fiber lengths from 3 to 30 m. The full lines are linear fits of the experimental points for each fiber. Fiber input average powers were (at 1 GHz): 420 to 1400 mW (3 m); 420 to 1360 mW (5 m); 200 to 1380 mW (10 m); 250 to 1300 mW (15 m); 200 to 1400 mW (30 m). (c) Theoretical $|{\beta }_2|/\gamma$ ratio (blue line) from simulation and experimental values of this ratio retrieved from measurement of the power and spectrum of the solitons, for several fiber lengths and soliton wavelengths. (d) Redshift of the soliton at maximum input power for varying fiber lengths. The dashed line is a guide to the eye.

Figure 3(b) shows the measured energy of the solitons through the whole tunability window. For each measured point in Fig. 3(b), we calculate the effective $|{\beta }_2|/\gamma$ ratio with Eq. (3), using duration values calculated from the Fourier transform of the measured spectrum. Figure 3(c) shows those experimental values along with the theoretical curve from finite element method calculations.

We find a good agreement between the simulation and the experiment for fibers up to 10 m, whereas in longer fibers, the calculated $|{\beta }_2|/\gamma$ ratios are lower than the theory. In a similar fashion, the highest redshift accessible via SSFS increases with fiber length, but it reaches a maximum value at $L=10\mathrm{m}$ [see Fig. 3(d)].

In fibers longer than 15 m, several mechanisms can explain the lower values of the effective $|{\beta }_2|/\gamma$ and the end of the redshift. First, the third-order dispersion arising when the soliton gets closer to the PBG edge decelerates the SSFS in the fiber.²⁸ Second, the attenuation experienced by the pulse when it propagates increases the nonlinear length $L_{\mathrm{NL}}=1/(\gamma P_0)$ which is the characteristic length needed for the soliton to adapt to any changes. At a certain point, the soliton can no longer follow the changes in $P_0$, ceases to redshift, and temporally broadens under dispersion. In the end, the redshifted pulse loses its soliton nature during propagation, and Eq. (3) is no longer valid. Figure 3(d) can be interpreted as the redshift versus the fiber length. In view of the above explanation, we can say that soliton nature of the pulse is retained until the plateau.

Finally, we achieve a 100 nm tuning range for the soliton. The length of the fiber must be less than 10 m so that Eq. (3) is still valid and the output pulse is Fourier-transform-limited. Then, the fiber length should be chosen according to the tunability and power requirements for the application. In the experiments described in the following, we used the 10-m fiber for spectroscopic application because it displays the best trade-off between tunability and soliton power, and the 3-m fiber for microscopy to get high soliton power.

3.2.

Microspectroscopy

We investigated the TPEF response of an aqueous solution of rhodamine 6G with the setup described above. We first compared the fluorescent levels and the signal-to-noise ratios in the cases of the excitation by solitons generated in the fiber at 80 MHz and at 1 GHz.

The TPEF signal at a given wavelength depends on pulse power, duration, and repetition rate $f_{\mathrm{rep}}$ in the following way:²⁹

More specifically, we expect the fluorescence signal to scale linearly with the repetition rate. If the TPEF photon count $S_{1\mathrm{sol}}$ arising from excitation with a single soliton has mean $\langle S_{1\mathrm{sol}}\rangle$ and standard deviation ${\sigma }_{1\mathrm{sol}}$, then it follows from the properties of the standard deviation that the TPEF count rate arising from a train of solitons at repetition rate $f_{\mathrm{rep}}$ has mean $\langle S\rangle \propto f_{\mathrm{rep}}\langle S_{1\mathrm{sol}}\rangle$ and standard deviation $\sigma \propto \sqrt{f_{\mathrm{rep}}}{\sigma }_{1\mathrm{sol}}$. We can then expect both the noise and the signal-to-noise ratio to scale with the square root of the repetition rate.

We measure a 10-fold increase of the fluorescence counts and a 4-fold increase of the signal-to-noise ratio under a 12.5-fold increase of the repetition rate (from 80 MHz to 1 GHz), as shown in the inset of Fig. 4. The experimental data seem to generally follow the trend predicted by the above calculation. This indicates that we observe only limited saturation effects in the sample when increasing the repetition rate.

Fig. 4

Two-photon excitation fluorescence (TPEF) spectra of a $170\mu \mathrm{M}$ solution of rhodamine 6G in water. Black squares: reference spectrum, measured by exciting the sample with the tunable fs light directly coming from the 80 MHz Ti:Sapphire laser (without passing through the fiber). Blue circles: spectrum acquired under excitation with solitons generated in a 10 m SC-PBG fiber pumped with an 800 nm, 1 GHz laser. Green triangles: spectrum acquired under excitation with solitons generated in a 10 m SC-PBG fiber pumped with an 800 nm, 80 MHz laser. Inset: TPEF traces under excitation with 870 nm solitons at 1 GHz (blue line) and 80 MHz (green line), with the corresponding signal-to-noise ratios (in ${\mathrm{cps}}^{1/2}$). Powers at the focus: black curve: 15 mW; blue: 5 mW; and green: 2 mW.

The main plot in Fig. 4 shows the TPEF spectra of the same rhodamine 6G solution. The two spectra (blue circles and green triangles) obtained with excitation by solitons at 80 MHz and 1 GHz were normalized with respect to their average power and repetition rate, taking into account the small saturation effect previously measured. The agreement between the two spectra is very good, proving the reliability of the soliton as a light source with only a little influence on the parameters of the pumping laser (spectrum, duration, chirp). Another spectrum was used as a control, using the fs pulse directly coming from the 80 MHz Ti:Sapphire laser (black squares) for excitation. The spectra show a good qualitative agreement. Our measurements are also consistent with the two-photon absorption cross section values for rhodamine 6G found in the literature.^29,30 However, no quantitative information can be retrieved from the comparison between this spectrum and two first spectra because many parameters vary (spectral width, pulse duration, beam shape).

3.3.

Microscopy

To demonstrate the imaging capabilities of our system, we performed images of several samples, including biological ones (see Fig. 5). We generated the solitons at 1 GHz repetition rate in a 3 m fiber to operate at high excitation power and low redshift (850 to 890 nm). We first tested the two epi-detection channels separately to obtain images of samples known for their strong TPEF and SHG signals, respectively. Figure 5(a) shows a TPEF image of a fixed Drosophila embryo, where the formaldehyde fixation process is responsible for inducing the fluorescence signal. Figure 5(b) shows the SHG image of ${\mathrm{BaTiO}}_3$ crystals. In both cases, the 50 to $70\mu \mathrm{s}$ dwell times that were used are comparable with the ones typically used in nonlinear microscopy. The multiphoton signal is about 10 times lower than images with standard Ti:Sapphire excitation. We have identified the reason for this discrepancy to be the spatial profile of the soliton that is singly peaked but not Gaussian. The microscope we used was not set up to compensate for this effect, which results in a bias against the soliton in this context. We then used the two channels to perform a multimodal image of a biological sample. Figure 5(c) shows a composite TPEF/SHG image of fixed mouse tumor skin tissue showing structural features that are not distinguishable in white light microscopy. The SHG emitting structures can be identified as collagen fibers, whereas TPEF comes from the stratum corneum.

Fig. 5

$300\times 300\text{pixels}$ TPEF and second-harmonic generation (SHG) images with 10 accumulations each. (a) TPEF image of a fixed Drosophila embryo obtained with an 870 nm soliton excitation and a $50\mu \mathrm{s}$ dwell time. (b) SHG image of ${\mathrm{BaTiO}}_3$ crystals obtained with an 850 nm soliton excitation and a $70\mu \mathrm{s}$ dwell time. (c) Composite image of fixed mouse tumor skin tissue obtained with an 870 nm soliton excitation and a $70\mu \mathrm{s}$ dwell time. TPEF is shown in red and SHG in green. Powers in the sample plane: (a) 40 mW; (b) 50 mW; and (c) 55 mW.