When a short (100-fs), transform-limited pulse in the 800-nm spectral region propagates in a single-mode fiber, normal dispersion induces a positive chirp in the pulse that exits the fiber. A grating pair may be used to precompensate the fiber dispersion, generating a negatively chirped pulse before it is injected into the fiber, so that in the linear regime (low power) a nearly transform-limited pulse may be obtained at the fiber output. (The pulse duration in the linear regime is limited in this case only by the presence of higher-order dispersion.) A number of applications are envisaged for fiber probes which involve the excitation of nonlinear processes by the probe, and thus require intense ultrashort pulses to appear at the exit face of the fiber. For example, in time-resolved near-field scanning optical microscopy (NSOM) pump-probe experiments, two short pulses with variable delay are propagated through a fiber with a near-field aperture at the output. The transient optical saturation of an absorbing medium in the near field of the fiber probe is measured, and the nonlinear signal is proportional to the square of the peak intensity at the fiber output [1
1. S. Smith, N. C. R. Holme, B. Orr, R. Kopelman, and T. B. Norris, “Ultrafast measurement in GaAs thin films using NSOM,” Ultramicroscopy 71, 213–223 (1998). [CrossRef]
]. Similarly, fiber probes have been used to excite two-photon fluorescence (TPF) of dyes, using either NSOM probes [2
2. M. Lewis, P. Wolanin, A. Gafni, and D. Steel, “Near-field scanning optical microscopy of single molecules by femtosecond two-photon excitation,” Opt. Lett. 23, 1111–1113 (1995). [CrossRef]
] or normal single-mode fiber [3
3. A. Lago, A. T. Obeidat, A. E. Kaplan, J. B. Khurgin, P. L. Shkilnikov, and M. D. Stern, “Two-photon-induced fluorescence of biological markers based on optical fibers,” Opt. Lett. 20, 2054–2056 (1995). [CrossRef] [PubMed]
]. In both cases, the fluorescence excitation is restricted to a region near the fiber output due to the I
dependence of the two-photon excitation coupled with the strong divergence of the beam [4
4. W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990). [CrossRef] [PubMed]
Because the signal in these applications involves nonlinear excitation of the system under study, one requires that the peak intensity of the pulse at the fiber output be maximized. This can be accomplished in the linear propagation regime by using the shortest possible pulses. At moderate to high powers, however, self-phase modulation (SPM) begins to modify the pulse spectrum and shape due to propagation through the fiber. In particular, for a negatively chirped pulse propagating in a normally dispersive fiber, SPM acts to narrow
the bandwidth of the pulse as it propagates, opposite the behavior of transform-limited or positively chirped pulses. In fact, a recent experiment demonstrated that with increasing fluence, the output pulse became broader yet remained close to the transform limit [5
5. B. R. Washburn, J. A. Buck, and S. E. Ralph, “Transform-limited spectral compression due to self-phase modulation in fibers,” Opt. Lett. 25, 445–447 (2000). [CrossRef]
The main consequence of the pulse broadening with increased input fluence is a reduction in the output peak intensity. Therefore the TPF will depend sub-quadratically with the input power to the fiber. In many situations, the average power of the pulse train in the fiber is fixed, e.g. by the requirement that the power delivered to the sample at the probe output be limited. For example, the average power delivered through metal-coated NSOM tips is usually limited to about 1 mW to avoid heating of the near-field probe. For fluorescence experiments in biological samples, the power must also be limited to avoid damaging the sample. For a fixed average power, the peak intensity can be varied by adjusting the laser repetition rate. In this paper, we consider the problem of maximizing the induced nonlinear signal in the presence of the average-power constraint. We find that the output pulse duration, and hence, the generated signal amplitude scale in a simple way with the input pulse, and discuss the consequences for the signal optimization.
We first consider the propagation of pre-chirped pulses in normally dispersive fiber. The laser used in the experiments was either a 90-fs Ti:sapphire mode-locked oscillator or 100-fs regenerative amplifier with a variable repetition rate up to 0.5 MHz [6
6. T. B. Norris, “Femtosecond pulse amplification at 250 kHz with a Ti:sapphire regenerative amplifier and application to continuum generation,” Opt. Lett. 17, 1009–1011 (1992). [CrossRef] [PubMed]
]. The initially transform-limited 800-nm pulses were doubled-passed through a 2000 ln/mm transmission grating [7
7. J.-K. Rhee, T. S. Sosnowski, T. B. Norris, J. A. Arns, and W. S. Colburn, “Chirped-pulse amplification of 85-fs pulses at 250 kHz with third-order dispersion compensation by use of holographic transmission gratings,” Opt. Lett. 19, 1550–1552 (1994). [CrossRef] [PubMed]
] to induce a negative chirp on the pulses to compensate for the positive chirp acquired in the fiber. The pulses were then coupled into a 90-cm long single-mode optical fiber (Newport F-SX) using a 10x microscope objective. The grating dispersion was adjusted to achieve the shortest pulse at the fiber output at low power levels. Representative autocorrelations and power spectra of the output pulse from the fiber at 0.16 nJ (solid line) and 0.4 nJ (dotted line) are shown in Fig. 1
. For input pulse energies below 20 pJ, the output pulse was as short as 170 fs, limited by uncorrected cubic and higher order phase error in the grating precompensator. Pulse broadening and spectral narrowing are evident at higher energies, similar to the results obtained by Washburn et al
. (Some spectral shift is also observed, which results from an asymmetry in the laser input pulse.) With our system, for pulse energies higher than about 4 nJ, the spectrum shows structure and the autocorrelation indicates the onset of pulse breakup, as illustrated in Fig. 2
. For the remainder of this paper, we consider principally energies below that where pulse breakup occurs.
Fig 1. (a) Autocorrelation width and (b) power spectra of output pulses from the fiber with energies of 0.16 nJ (solid line) and 0.4 nJ (dotted line)
Fig 2. (a) Autocorrelation and (b) spectrum of an output pulse with an energy of 8 nJ, illustrating the effects of pulse break up
Maintaining the same input chirp parameters (fixed grating positions), we then measured the intensity autocorrelation versus the average pump power to find the output pulse width dependence on input pulse energy. The results are shown in Fig. 3
(a). We find that, for fluences below that where pulse breakup begins to occur, the dependence can be fit reasonably well by a simple scaling law: τp
, where Up
is the fluence, τp
is the output pulse duration, τpo
is the output pulse duration at low power, β
is a proportionality constant, and α=0.75 (±0.05). We modeled the propagation of negatively chirped pulses in the fiber with the nonlinear Schrödinger equation using the split-step method. The model included SPM and second- and third-order dispersion only. We found that for a given pulse shape, the pulse width followed a power law dependence but the actual scaling coefficient depended on the pre-chirp and pulse shape. For reasonable parameters, a similar scaling behavior was found.
Here, we consider two limiting cases. In the low pump power limit where the pulse width is independent of the pump fluence, Ufl
, and if R is the repetition rate of the amplifier, the average fluorescence power Wfl
·R will depend quadratically on the average seed power: Wfl
. In the limit where nonlinear propagation effects give rise to a significant change in the pulse width such that Δτp
, we find Ufl
. In terms of average power, in this limit, we have Wfl
1-α. For a fixed repetition rate at high fluences, we thus expect a simple power dependence of the TPF signal with γ=2-α.
In order to investigate the scaling dependence of the TPF signal, we used two-photon excitation of Rhodamine 6G in methanol. We first verified the quadratic dependence Wfl
in the absence of propagation effects by focussing the laser directly into the dye solution (i.e. bypassing the grating stretcher and fiber); we found γ
=2.2±0.2. (Note that this implies the absence of pump depletion or photobleaching effects.) We then coupled the stretcher output into the fiber and dipped the fiber into the dye solution. The TPF excited at the exit tip of the fiber was imaged onto a CCD. The experimental TPF signal versus pump power is shown in Fig. 3
(b). The solid line is plotted using Eq. (1)
where the parameters, τpo
, and β
are determined from the autocorrelation experiment (Fig. 3
(a)). The power dependences of pulse width and fluorescence power are consistent with our expectations within experimental error.
Fig 3. Pump power (energy) dependence of (a) autocorrelation width and (b) TPF signal amplitude
If the average pump power is limited to some maximum value, then one may consider the scaling of the TPF signal versus repetition rate R at fixed seed power. In the low power limit where the pulse width is independent of pump fluence, Wfl
∝1/R. In the higher power limit where Δτp
, we see that Wfl
α-1. In either case, the total nonlinear signal can be increased by reducing the repetition rate as long as the pulse width scales sub-linearly with fluence, i.e. α<1. At fixed average power, a lower repetition rate gives a higher pump fluence. Nonlinear propagation effects cause the pulse width to increase, but since the TPF increases with the square of the peak intensity, it can be maximized by increasing the energy per pulse as long as the pulse width increases sub-linearly with pulse fluence. Of course this conclusion is valid only as long as the pulse scaling behavior is valid; at sufficiently low repetition rate and fixed average power, the pulse fluence will be so high that the pulse begins to break up, generally resulting in substantially reduced peak power and hence decreasing TPF signal.
In Fig. 4
we show the TPF signal as a function of repetition rate using the regenerative amplifier system, at a fixed average power of (a) 0.3 and (b) 0.9 mW. As evident in Fig 4
(a), the TPF signal clearly increases with decreasing repetition rate, until the scaling relation breaks down at approximately R=75 kHz (corresponding to a pulse energy of 4 nJ). However, even in the range of energies between 4 nJ and 8 nJ where we cannot apply the simple scaling behavior, the pulse width still changes sub-linearly with the pump energy and the TPF signal continue to increase for lower R. In the higher average power case, we can see that the rate of change of the TPF signal with the pump power is much slower, and the optimal repetition rate after which the TPF signal does not change appreciably with pump power is higher. Of course, since U
, if higher average power is allowed, then the optimal repetition rate will be higher; this trend is apparent in Fig. 4
Fig 4. Repetition rate dependence of TPF signal amplitude at fixed average powers of (a) 0.3 mW and (b) 0.9 mW
In summary, we have studied the propagation of negatively pre-chirped pulses in a normally dispersive fiber, and the excitation of nonlinear processes at the fiber output. We have observed that for energies below that at which pulse breakup begins to occur, the pulse width scales sublinearly with energy. This results in a subquadratic scaling of TPF or other nonlinear processes with pump power. For fixed pump average power, this scaling implies that the nonlinear signal can be optimized by operating at a repetition rate with a corresponding energy which is just below the energy at which pulse distortion becomes significant.