... tcole Polytechnique de Montreal, P.O. Box 6079 Station A, Montreal, Qu6bec H3C ... Center for Research in Electro-Optics and Lasers, University of Central ...
1996
OPTICS LETTERS / Vol. 18, No. 23 / December 1, 1993
Enhanced self-phase modulation in tapered fibers P. Dumais, F. Gonthier, S. Lacroix, and J. Bures Department of Engineering Physics, tcole Polytechnique de Montreal, P.O. Box 6079 Station A, Montreal, Qu6bec H3C 3A7, Canada
A. Villeneuve, P. G. J. Wigley, and G. I. Stegeman Center for Research in Electro-Optics and Lasers, University of Central Florida, 12424 Research Parkway, Orlando, Florida 32826 Received April 30, 1993
Fiber tapering reduces the effective area and increases the Kerr nonlinear phase shift of the fundamental mode. A significant nonlinear effect was observed in a micrometer-diameter centimeter-long section of tapered fiber at a wavelength of 1550 nm. The numerical simulation of a second-order soliton entering the taper gives a spectral widening that matches the observed spectrum.
In silica optical fiber the nonlinear Kerr coefficient is very small (n2 = 3.2 X 10-20 m 2/W), and observation of a power-dependent phase shift requires high intensities or long interaction distances. Fiber tapering increases light concentration and therefore can generate the same phase shift on much shorter distances for a given power. In this Letter we present some experiment results involving the injection of 350-fs pulses of 1550-nm wavelength at soliton-generating intensities in a single-mode optical fiber containing a taper and carry out the theoretical analysis and numerical simulation reproducing these results. This experiment is the first step in the making of a pulsedriven switching device. We begin with the theory leading to the calculation of the nonlinear phase shift. It shows how a perturbative scalar approximation may be used to estimate the variation of the propagation constant with regard to power and hence the resulting nonlinear phase shift. We calculate numerically the taper's propagation constant for the fundamental mode and the corresponding field distribution. The calculations are based on a scalar shooting method, which has advantages over analytical calculation of the modes of a three-layered waveguide. It permits us to use the exact or measured refractive-index profile of the fiber and to make intensity-dependent corrections on it. We can calculate by iterative corrections the nonlinear modes of a tapered fiber. All calculations presented here are made with the measured profile, showing a dip in the center, of Corning SMF-28 fiber. The scalar perturbative calculation' yields the correction to the propagation constant of the fundamental mode
k it(n
A13= P/-P =
- -fi);f2d
AXf~ n~~ 7 E)P~ d A ]
(. JA.
2
AA Ir Uzi
It
1000
kn 2P Aeff
mode (or of any field distribution).
100
10
(1)
where T is the scalar field, A. is the infinite cross section, k is the wave number (27r/A), and Aeff = 2 / fA,,L T 4 dA is the effective area of the [fA. iT2dA] fundamental
is the fiber cross section, which contains the nonlinearity. Symbols with overbars represent unperturbed, i.e., linear, quantities. We compared Eq. (1) with the variation of the propagation constant calculated with our corrective shooting method, subtracting the linear propagation constant from the nonlinear one. We found that the use of this equation introduced only a small error, which is highest for small diameters, reaching 10% for a diameter of 0.8 jam. For a peak power of 1 kW, the variation of the propagation constant can be considered as a perturbation. Figure 1 shows the effective area of the fundamental mode LP01 for the fiber used as a function of taper diameter. Decreasing the effective area increases the intensity P/Aeff and the nonlinear phase shift A,8L, where L is the propagation length. We made the tapers by pulling a fiber heated by a large (6-mm) flame and controlled the diameter by monitoring the beating between the LPo1 and LP11 modes excited at 633 nm.2 Figure 1 relates the evolution of beam confinement throughout fiber
ANL
0146-9592/93/231996-03$6.00/0
0.1
Diameter (pam)
Fig. 1. Computed effective area of the fundamental mode of Corning SMF-28 fiber as a function of fiber diameter at A = 1550 nm. i 1993 Optical Society of America
December 1, 1993 / Vol. 18, No. 23 / OPTICS LETTERS
tapering.
At full diameter the power is guided by
the fiber core, and the effective area of 88
/Lm
2
is of
the same order of magnitude as the geometric area of the core. As the fiber is stretched, the core area decreases, as does the effective area, until the mode enters the core-cladding interface. Then the power escapes form the core and is guided by the cladding. From this point on, the core has little effect, and the effective area is identical to that
of a single-
medium fiber of the same diameter. Finally, the effective area reaches a minimum, after which the power starts to leak away from the cladding into the surrounding medium. There are two minima on the curve in Fig. 1. The one occurring at the larger diameter (--100 jnm) corresponds to maximum confinement in the core. The other (at 0.8 jim) corresponds to maximum confinement in the domain in which the light is guided by the cladding. A taper 0.8 jum in diameter leads to an effective area that is -80 times smaller than the effective area for an untapered fiber (125-jim diameter). This is the optimum confinement found for a taper of any size for the given fiber at a 1550-nm wavelength.
Thus a taper of this diameter should maximize the magnitude of the nonlinear Kerr effect. The fiber used was Corning SMF-28, a standard telecommunication fiber. The minimum diameter of the taper was 1.1 jim throughout a length of 9 mm, and the leading and trailing sections were smooth enough to prevent transfer of power between guided modes of the taper. 3
We measured 10% loss of power
within the taper at 1550-nm wavelengths, using cw power not exceeding 1 mW (linear regime). Tapers of smaller diameter are difficult to realize and tend to be much lossier because of surface scattering. This is why we used 1.1-jim diameter instead of the computed optimum of 0.8 jim. The taper was preceded by 1 m and followed by 10 cm of untapered
fiber. The ultrashort light pulses injected into the fiber were generated by a color-center laser pumped by a frequency-doubled YLF laser employing an additivepulse mode-locking technique.
As much as 80 mW
of average power at 1550 nm was available in 350-fs at 76 MHz. For the data presented pulses (TFWHM) here, the average power transmitted through the
1997
results cannot be reproduced by this simplified analysis. Indeed, estimation of the self-phasemodulation bandwidth remains the same without regard to the position of the taper within the length of untapered fiber. However, experiments showed that variation of the input length of untapered fiber preceding the taper had a dramatic effect on the spectral broadening, whereas modification of the length of fiber following the taper had little effect. To replicate the measured spectrum with a numerical simulation, we needed to propagate a second-order soliton in a fiber of diameter and characteristics comparable with those of the sample used. The profile of the taper was computed and then approximated by a cylindrical section of 1.1-jim diameter and 9-mm length surrounded by conic sections with exponential slopes. In the simplified nonlinear propagation equation4 2
aA a A iy-/=282
2A, yJAI
(2)
two terms are diameter dependent: the dispersion /62 and the nonlinear parameter y = 27rn2/AAeff, which varies as the nonlinear phase shift and hence follows the inverse of the effective area of Fig. 1. The dispersion, defined as /32
A2 ±
(2,rC)2 (
=
(3)
A)
was calculated as a function of the fiber diameter (see Fig. 3) in the same manner as we did for the effective area. For a given diameter, we calculated the propagation constant for a few wavelengths around 1550 nm and the derivatives in Eq. (3), using finite differences. For very large diameters, for which the waveguide boundaries are distant enough to be neglected, the dispersion approaches the material dispersion (28 ps2 /km). The dispersion varies most where the field is affected by waveguide boundaries, as we have seen for the effective area. The small maximum in the curve of Fig. 3 appears at the area at which the field escapes from the (germanium-doped) core into
sample was 30 mW, corresponding to 1.1 kW of peak
power. This peak power corresponded closely to that of a second-order soliton in our fiber.4 Comparison of the input pulse with a simulated second-order soliton pulse of 350-fs duration shows that the laser pulses are almost transform limited. The spectral width can be expressed by AvTo in normalized units, 7 6 . The where v is the frequency and To = TFWHM/1. spectral width of the input pulse is 0.3 normalized units. The pulse spectrum measured from this experiment at the output of the fiber is shown by the
1.6
1.2
I Iz9
0.8
0.4
solid curve in Fig. 2.
The resultant
self-phase-modulation estimated
from Eq. (1) yields a maximum
phase
shift of w
through the fiber-taper combination, with contributions from both the fiber and the taper being comparable. This should manifest itself as a simple spectral broadening.4 However, the experimental
-3
-1
0 1 (V- VdTO
2
3
Fig. 2. Measured (solid curve) and simulated (dashed curve) output spectra.
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OPTICS LETTERS / Vol. 18, No. 23 / December 1, 1993 400
300
200
A0
100
0
~
*1 0 0
0.1
_
_
1
_
_
_
10
_
_
100
_
1000
Dimeter (pm)
Fig. 3. Computed dispersion of the fundamental A = 1550 nm in Corning SMF-28 fiber.
at
the cladding, a material with the same dispersion characteristics. The largest variation occurs at the smallest diameters in Fig. 3 as the field penetrates the air, tending to a much larger variation in the refractive index and hence a significant change in the dispersion. Note that, if the curve in Fig. 3 were plotted for smaller diameters, the dispersion should decrease again and tend to the material dispersion of air. In our calculations the refractive index of air is set to unity, and its dispersion would be null. The actual dispersion of air is a few orders of magnitude lower than that of silica, so this is a good approximation. Incorporation of the data for both effective area and dispersion in a beam-propagation-methodlike algorithm was straightforward. In an untapered fiber, the second-order soliton passes through a cycle of a single contraction and reconstruction that, of course, lasts for a soliton period. The soliton period for our fiber and for the pulses used was 2.75 m. At 1 m from the entry face, contraction of the pulse was clearly present, and, if the pulse entered the taper at this stage, the resultant spectral broadening was enhanced. The spectrum generated from this simulation under these conditions is shown by the dashed curve in Fig. 2. Further propagation of this pulse in the untapered portion ofthe fiber brings little modification to its spectrum. We compare in Fig. 2 both experimental and simulated spectra, obtained as described. Note that Kuehl5 predicted that the spectral broadening of a second-order soliton could be enhanced by fiber tapering. In this experiment ultrashort pulses were injected into a taper made of some length of standard fiber, and the spectral broadening was observed. The untapered ends of the sample were then shortened to isolate the effect of the taper alone, and com-
parison among spectra obtained for different fiber lengths showed that the spectral broadening varied noticeably with the taper position within the fiber, illustrating the inadequacy of the self-phasemodulation estimation. We used soliton compression in a simulation to reproduce the spectral broadening that was observed. It can seem from Fig. 2 that the width of both experimental and computed spectra measures approximately 6 normalized units, corresponding to 250 nm. The general topology is also the same, although the experimental spectrum manifests some asymmetric broadening that could be explained by stimulated Raman scattering6' 7 or by an asymmetrical input pulse, neither of which was included in this preliminary simulation. Overall, the simulated spectra are in good agreement with the essential features measured. None of the tapers used in the experiment, the one described and similar manufactured tapers, was destroyed by optical breakdown at intensities estimated at 1015 W/m2 . This is encouraging for the use of tapers in optical switches relying on nonlinear phase change. Using the data for dispersion and effective area, we see that tapers can support solitons, and therefore a solitonic switching device consisting of a taper in one of the branches of a Mach-Zehnder fiber interferometer can be constructed. Also, a double-pass geometry achieved with a Michelson configuration could halve the taper length needed. Such a device, which uses the taper to generate a uniform nonlinear phase shift, would permit the complete switching of a pulse injected into it and is a promising avenue for all-fiber optical switches. Funds for this research at l'Ecole Polytechnique de Montr6al were provided by the Natural Sciences and Engineering Research Council of Canada, grant OGP0008379, and Coop6ration Internationale, projet Qu6bec/Etats-Unis Q/EU-90-05-0011. The research at the Center for Research in Electro-Optics and Lasers was sponsored by the National Science Foundation.
References 1. A. W. Synder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 18, p. 376. 2. F. Gonthier, J. Lapierre, C. Veilleux, S. Lacroix, and J. Bures, Appl. Opt. 26, 444 (1987). 3. J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S.
Lacroix, and F. Gonthier, Proc. Inst. Electr. Eng. Part J 138, 343 (1991).
4. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989), Chap. 2, p. 43. 5. H. H. Kuehl, J. Opt. Soc. Am. B 5, 709 (1988).
6. J. P. Gordon, Opt. Lett. 11, 662 (1986). 7. F. M. Mitschke and L. F. Mollenauer, Opt. Lett. 11, 659 (1986).
