It is proposed for the first time how an electrically poled chirped Bragg grating in an optical fiber may be used for creating efficient, electrically controllable optical fiber delay lines. The tunable range for the delay can be chosen by fixing the chirp rate of the Bragg grating. Additionally, strain and temperature may be used to change the centre wavelength of the delay line. Dispersion may also be used to increase phase-modulation in nonlinear optical fibers.
©2006 Optical Society of America
Increasing the optical nonlinearity in optical fibers is of great interest and has been the subject of study over many years [1–3]. For this purpose, several glass compositions  and taperedfibers  have been researched. Recently, how the effective refractive index of the mode can enhance the nonlinearity of nano-wires has been reported . It is therefore of interest if standard optical fibers could be used with an enhanced nonlinear effect. In this paper we propose a simple technique to enhance the nonlinear effect in electro-optically poled optical fibers for optical delay lines, with implications for many other applications.
Optical delay lines are of great importance in a variety of applications such as in optical communication and for controlling interferometers in sensors. Commercially available delay lines are made with mirrors mounted on a translation stage, or by stretching or by heating  optical fibers. Tuning the reflection wavelength by stretching [6–7] of fibre Bragg gratings is very useful as it allows the Bragg wavelength to be changed over a large value (>1%). However, this mechanical technique is usually slow. Alternatively, mechanically switching-in of discrete sections of in-line optical fiber can also change the optical delay in steps. Large values of delay require long path length changes; additionally, tuning the delay is generally slow as it has to be performed mechanically. Silica fiber has a refractive index around 50% higher than air, which makes it suitable for fabricating more compact delay lines compared to free-space versions. If the fiber is made electro-optic, the delay can be changed electrooptically. Optical fibers have been shown to have an induced electro-optic coefficient once they are poled . This process requires an optical fiber core to be subjected to a high voltage at an elevated temperature of ~260 °C and allowed to cool with the field left on; this process renders the fiber electro-optic . Subsequently, the refractive index of the fiber (and hence the phase of propagating light) can be altered by the application of an electric field. However, considering that a maximum induced electro-optic coefficient is only ~1pm/V, the maximum phase change of ~100π at a wavelength of 1.55 microns can be expected in a 500 mm length of poled fiber. This is equivalent to a refractive index change of ~0.5×10-4 for a reported nonlinearity of 0.4pm/V . Assuming that the final useable nonlinearity can be 1pm/V, this translates to an ~50 micron path length change, equivalent to an optical delay change of 0.25 ps, far too small to be of much use for applications requiring 10’s of ps of delay. The electrooptical effect in optical fibres with Bragg gratings would induce a very small shift in the Bragg wavelength (~0.2nm, equivalent to 0.006% at 1550nm, as discussed later). It would be very useful if the effect of the small nonlinearity could be increased as there is need for a large electro-optically tunable delay, and that is proposed for the first time in this paper.
The optical delay through an optical fiber is given by t=ngL/c, where ng is the effective group index of a mode in the optical fiber, L is its length and c is the velocity of light. Using the standard values for single-mode optical fiber, a commonly used figure for the delay through an optical fiber is ~5ns/m. We consider here an electrically poled section of optical fiber. The change in the delay in the poled optical fiber as a function of the change in the effective group index (≈ change in the effective index) which is dependent on the applied voltage, V, is:
Thus the change in delay through the optical fiber is directly proportional to and linearly dependent on the change in the effective index of the fiber mode. The best that can be achieved is linearly proportional to the length over which the refractive index is altered. If we restrict the discussion to compact devices of length ~100mm, the maximum delay change is too small to be useful.
In an optical fiber with a chirped fiber Bragg grating of length L, the change in delay at a wavelength detuning, δλB within the chirp bandwidth, Δλ is:
The Bragg wavelength, λB is :
where Λlocal is the local period of the grating and neff is the effective index of the mode. In a poled fiber, the Bragg wavelength changes as a function of the refractive index as:
Therefore using Eq. (2),
Now 2t/Δλ is the dispersion, D in ps/nm, and defining the central Bragg wavelength of the chirped Bragg grating as λB, we arrive at:
where, we have assumed that the Bragg wavelength is more-or-less constant. Equation (6) demonstrates how the nonlinearity of the electro-optic fiber is suddenly magnified by the dispersion parameter, D. This relationship has a profound influence on the effective nonlinearity of nonlinear systems, as the effective nonlinearity is significantly modified by dispersion and may be used to benefit delay lines and other electro-optic or nonlinear devices.
In order to understand the significance of Eq. (6) we relate the change in refractive index to the electro-optic coefficient, r33 in which the input polarization and the applied field are in the same transverse direction ,
where r33 is the induced electro-optic coefficient in the poled fiber, V the applied voltage, and d the distance between the electrodes. We therefore introduce the concept of dispersion enhanced effective nonlinearity of a poled fiber as:
Equation (8) clearly demonstrates how dispersion in the poled fiber fundamentally changes the electro-optically induced delay.
3. Results of simulation
Figure 1 shows the electro-optically induced delay for 100mm long chirped fiber gratings of different bandwidths as a function of the refractive index change by the application of an electric field. The Bragg wavelength shifts by ~0.2nm per 10-4 change in the effective index of the mode. For a refractive index change of 5×10-5 (~ half the maximum possible for a poled fibre ), the induced delay is 1500ps for a grating with a chirp bandwidth of 0.1nm. This value is nearly 6,000 times larger than the delay induced in transmission for a 500mm long poled fibre, and is limited by the bandwidth of the grating, as the electro-optically induced shift in the central wavelength is equal to the bandwidth of the chirp. For the largest chirp bandwidth considered in Fig. 1 (0.8nm) the maximum delay is ~200ps for an index change of 10-4, significant for such a short device. Thus it is clear that the optical delay may be electro-optically tuned using a chirped grating, however over a limited bandwidth. With longer lengths of poled chirped gratings, one can change the bandwidth to arrive at any desired value of optical delay. Significantly, reversing the sign of the chirp or the voltage reverses the sign of the electro-optically induced optical delay by the same magnitude.
Finally, re-formulating Eq. (8) into a phase change for the reflected light as a function of applied voltage at the operating Bragg wavelength, we arrive at:
Equation (9) allows the calculation of the electro-optically induced phase change. Figure 1 also shows the phase-change for the various grating parameters indicated on the graph. The phase change for any grating can also be derived from Eq. (6) as:
which is ~150,000π phase-change when the refractive index alters by 10-4 for a 100mm grating with a bandwidth of 0.2nm.
In order to tune the operating wavelength of the device, strain or temperature may be used to set the wavelength . The device can be used by positioning the operating wavelength at the centre or at one end of the chirped grating bandwidth. Applying a positive or negative voltage will either increase or reduce the delay of the reflected wavelength.
In Fig. 2, is shown the group delay and reflection spectra of a 100mm long apodised grating with a chirp bandwidth of 0.8nm. The reflection spectra and the delay change on applying a voltage to the electro-optic fibre are also shown. The shift in the wavelength of Bragg reflection is 0.2nm and the change in the delay is ~300ps for a change in the effective index of the mode of 10-4. It is clear that this technique may be applied to poled super-structure gratings, or in other electro- optical materials such as LiNbO3. Indeed, using metal electrodes in poled fiber  with a chirped grating to heat it may also be used as an effective method of tuning the delay by a larger amount than is possible using the electro-optic effect in optical fibers, albeit at a slower speed, providing a tunable delay for multi-channel applications and in signal processing.
It is clear that changing the effective index of a mode in the optical fiber by whichever means (optical nonlinearly, strain or temperature induced) will have the same effect if the fiber has a large dispersion, such as with a chirped Bragg grating. Thus the small change in the refractive index using a high intensity optical pulse will change the group delay of another within the band-gap of the grating, through cross-phase modulation. Whatever the source maybe for the induced index change, δneff, the delay and the phase will be magnified as per Eqs. (6) and (10) respectively.
4. Optical pulsed sources
As the electro-optic effect in poled fibres has been demonstrated to operate at MHz frequencies , the electro-optically tuneable delay line could serve as an excellent device for use in applications such as optical coherence tomography (OCT) in which a non-mechanical, fast delay line would be of great benefit for large depth of field and fast acquisition time (>MHz). This would of course require longer gratings with a much wider bandwidth to accommodate the broader bandwidth of OCT sources (50nm). For example a 10 meter grating with a 100 nm chirp bandwidth would accommodate a source with a 96.8nm bandwidth to provide a delay change of ~±200ps (±20 mm). In several earlier papers, we have demonstrated the dispersion compensation of pulses using back-to-back chirped FBGs [13–15]. Recently , it was also demonstrated that the dispersion in chirped grating delay lines may be partially removed by using two concatenated chirped gratings back-to-back. In the latter scheme, the delay is changed by straining one of the chirped FBGs which changes the Bragg wavelength  and dispersion compensated by a second grating back-to-back.
We propose a novel modification which not only allows the dispersion to be compensated but also allows the delay to be doubled or increased by a larger factor electro-optically, by concatenating several pairs of gratings in the fashion shown schematically in figure 3. For femtosecond pulses, figure 3 shows how a voltage tuneable delay may be induced by using two gratings; one to tune the delay, and the second identical chirped grating in reverse, to compensate for the dispersion. However, if both gratings are poled, the delay is doubled and the dispersion compensated at the same time. Indeed, if dispersions in the two gratings are not identical, additional pulse shaping may also be possible.
We further consider the action of dispersion compensation for pulses so that only the delay is observed. As we have reported earlier, femto-second pulses may be dispersed in optical fibres and fully recovered using chirped gratings . Any uncompensated part can appear as a dispersed background, adding to noise and additional structure . The tuning of the gratings should not add additional distortion unless the group delay is non-linear with respect to wavelength. This is usually the case in strongly reflecting gratings  and is a function of the form of apodisation as well. A good delay line will therefore require a very smooth and linear dispersion, and for this application well apodised chirped gratings with modest reflectivities (R~ 80%) are recommended.
In conclusion we have theoretically proposed for the first time how a dispersive element such as a chirped grating in a poled optical fiber may be used as an electro-optically tuneable delay line. It has been shown how the effect of the electro-optic nonlinear coefficient is dramatically modified by the introduction of dispersion. This fact has significant consequences for a number of optical waveguide devices, not only restricted to electro-optic devices, but also to other nonlinear optical effects. We are currently investigating the fabrication of such devices in our laboratory.
The author acknowledges the Canada Research Chairs program of the Natural Science & Engineering Research Council of Canada (NSERC) for their support of the research.
References and links
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