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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 19 — Sep. 13, 2010
  • pp: 19844–19859
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Point-by-point written fiber-Bragg gratings and their application in complex grating designs

Graham D. Marshall, Robert J. Williams, Nemanja Jovanovic, M. J. Steel, and Michael J. Withford  »View Author Affiliations


Optics Express, Vol. 18, Issue 19, pp. 19844-19859 (2010)
http://dx.doi.org/10.1364/OE.18.019844


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Abstract

The point-by-point technique of fabricating fibre-Bragg gratings using an ultrafast laser enables complete control of the position of each index modification that comprises the grating. By tailoring the local phase, amplitude and spacing of the grating’s refractive index modulations it is possible to create gratings with complex transmission and reflection spectra. We report a series of grating structures that were realized by exploiting these flexibilities. Such structures include gratings with controlled bandwidth, and amplitude- and phase-modulated sampled (or superstructured) gratings. A model based on coupled-mode theory provides important insights into the manufacture of such gratings. Our approach offers a quick and easy method of producing complex, non-uniform grating structures in both fibres and other mono-mode waveguiding structures.

© 2010 OSA

1. Introduction

Fibre Bragg gratings (FBGs) inscribed with femtosecond lasers are of great interest in applications ranging from cavity mirrors in high powered fibre lasers to sensing in harsh environments [1

1. D. Grobnic, S. J. Mihailov, C. W. Smelser, and H. M. Ding, “Sapphire fiber Bragg grating sensor made using femtosecond laser radiation for ultrahigh temperature applications,” IEEE Photon. Technol. Lett. 16(11), 2505–2507 (2004), http://dx.doi.org/10.1109/Lpt.2004.834920. [CrossRef]

,2

2. N. Jovanovic, M. Åslund, A. Fuerbach, S. D. Jackson, G. D. Marshall, and M. J. Withford, “Narrow linewidth, 100 W cw Yb3+-doped silica fiber laser with a point-by-point Bragg grating inscribed directly into the active core,” Opt. Lett. 32(19), 2804–2806 (2007), http://dx.doi.org/10.1364/OL.32.002804. [CrossRef] [PubMed]

]. The unique properties of such gratings stem from their method of manufacture which differs from the more common mode of UV laser and phase mask fabrication. The most notable difference is the fact that a femtosecond laser interacts with a dielectric material via nonlinear photoionization mechanisms such as multiphoton and tunnelling ionization; this removes the need for glass photosensitivity at the wavelength of the writing laser [3

3. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008), http://dx.doi.org/10.1038/nphoton.2008.47. [CrossRef]

]. Gratings can therefore be inscribed directly into a wide range of dielectric materials including rare-earth doped fibres used in fibre laser applications [4

4. Y. Lai, A. Martinez, I. Khrushchev, and I. Bennion, “Distributed Bragg reflector fiber laser fabricated by femtosecond laser inscription,” Opt. Lett. 31(11), 1672–1674 (2006), http://dx.doi.org/10.1364/OL.31.001672. [CrossRef] [PubMed]

,5

5. E. Wikszak, J. Thomas, J. Burghoff, B. Ortaç, J. Limpert, S. Nolte, U. Fuchs, and A. Tünnermann, “Erbium fiber laser based on intracore femtosecond-written fiber Bragg grating,” Opt. Lett. 31(16), 2390–2392 (2006), http://dx.doi.org/10.1364/OL.31.002390. [CrossRef] [PubMed]

]. Other advantages include strong polarization-dependent properties [6

6. N. Jovanovic, J. Thomas, R. J. Williams, M. J. Steel, G. D. Marshall, A. Fuerbach, S. Nolte, A. Tünnermann, and M. J. Withford, “Polarization-dependent effects in point-by-point fiber Bragg gratings enable simple, linearly polarized fiber lasers,” Opt. Express 17(8), 6082–6095 (2009), http://dx.doi.org/10.1364/OE.17.006082. [CrossRef] [PubMed]

] (caused by the ellipsoidal morphology of the grating’s periods) and enhanced grating stability and resilience to heating to temperatures approaching the softening point of the glass (typically > 1000°C [7

7. D. Grobnic, C. W. Smelser, S. J. Mihailov, and R. B. Walker, “Long-term thermal stability tests at 1000°C of silica fibre Bragg gratings made with ultrafast laser radiation,” Meas. Sci. Technol. 17(5), 1009–1013 (2006), http://dx.doi.org/10.1088/0957-0233/17/5/S12. [CrossRef]

,8

8. A. Martinez, I. Y. Khrushchev, and I. Bennion, “Thermal properties of fibre Bragg gratings inscribed point-by-point by infrared femtosecond laser,” Electron. Lett. 41(4), 176–178 (2005), http://dx.doi.org/10.1049/El:20057898. [CrossRef]

]).

Two techniques have been demonstrated for grating inscription using ultrafast lasers, namely the point-by-point (PbP) and phase mask scanning techniques [9

9. A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, “Direct writing of fibre Bragg gratings by femtosecond laser,” Electron. Lett. 40(19), 1170–1172 (2004), http://dx.doi.org/10.1049/El:20046050. [CrossRef]

,10

10. S. J. Mihailov, C. W. Smelser, P. Lu, R. B. Walker, D. Grobnic, H. M. Ding, G. Henderson, and J. Unruh, “Fiber bragg gratings made with a phase mask and 800-nm femtosecond radiation,” Opt. Lett. 28(12), 995–997 (2003), http://dx.doi.org/10.1364/OL.28.000995. [CrossRef] [PubMed]

]. When a laser is focused through a phase mask and into a fibre held proximal to the mask, the interference pattern generated by the mask can be inscribed into the fibre core. By translating the fibre and phase mask simultaneously with respect to the writing laser beam it is possible to inscribe index modifications that extend throughout the whole guided mode region [11

11. D. Grobnic, C. W. Smelser, S. J. Mihailov, R. B. Walker, and P. Lu, “Fiber Bragg gratings with suppressed cladding modes made in SMF-28 with a femtosecond IR laser and a phase mask,” IEEE Photon. Technol. Lett. 16(8), 1864–1866 (2004), http://dx.doi.org/10.1109/LPT.2004.831239. [CrossRef]

] and that may be as long as the phase mask [12

12. J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber Bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys., A Mater. Sci. Process. 86(2), 153–157 (2006), http://dx.doi.org/10.1007/s00339-006-3754-2. [CrossRef]

]. The principal limitation of this phase mask technique is that only a single Bragg period can be obtained from any given mask. The creation of complex grating structures requires complex phase masks which can be expensive to manufacture.

2. Experimental setup

In order to fabricate the FBGs described herein, the polymer jacket was chemically stripped from standard telecommunications SMF-28e optical fibre and the fibre positioned at the focus of a high numerical aperture focusing lens. In order to maintain consistent alignment of the fibre with the laser focus, the fibre was threaded through a glass ferrule and held in front of a 0.8 N.A., 20× oil immersion objective lens. The ferrule (and therefore the centre of the fibre core) was precisely positioned relative to the lens using a 3-axis flexure translation stage which provided additional piezo-control and thus sub-micron fidelity in the positioning of the laser focus relative to the fibre core. A schematic of the system is shown in Fig. 1
Fig. 1 (a) Schematic of the FBG writing setup showing focusing objective, and optical fibre being held in the alignment ferrule. For clarity the working distance of the objective is exaggerated and the objective’s immersion oil is not shown. (b) A micrograph of the end of the 1 mm diameter (10 mm long) ferrule showing the D-shaped cross-section. The ferrule was sourced from Thorlabs (part no. TS125) and polished in-house to achieve the D-shaped section.
. The output face of the objective lens, the fibre and the ferrule were all immersed in index matching oil. A high-precision air-bearing translation stage was used to draw the fibre through the ferrule at a constant velocity while single femtosecond laser pulses inscribed each refractive index modulation of the grating. Fibre gratings were written with pulse energies that ranged between 120 and 250 nJ; the laser pulses of duration < 120 fs were generated by a regeneratively-amplified Ti:Sapphire femtosecond laser operating at 800 nm and with a typical 1 kHz repetition rate. Second-order gratings with target Bragg wavelengths λB in the range 1520 - 1570 nm (corresponding to translation speeds of 1.05 – 1.09 mm/s respectively) were inscribed. The gratings were analysed during fabrication using a broadband EE-LED source and a high resolution (10 pm) optical spectrum analyzer (Advantest Q8384) or using a swept wavelength system (JDSU 15100) which, in combination with an optical circulator, provided higher resolution (3 pm) and sensitivity (>50 dB SNR) reflection and transmission data.

The central Bragg wavelength of the inscribed gratings is determined by a modified version of the standard Bragg equation that includes the order of the grating m, the velocity of the fibre drawing stage v and the pulse repetition frequency of the laser f

λB=2neffvmf.
(1)

3. Bandwidth and coupling constant control

R=tanh2(κL).
(2)

The coupling coefficient depends on the refractive index modulation Δn and the Bragg wavelength

κ=πλBΔn.
(3)

The width of a grating’s resonance (defined as the wavelength interval between the first zeros in the reflection spectrum) depends [18

18. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997), http://dx.doi.org/10.1109/50.618322. [CrossRef]

] on both the grating coupling constant and length

Δλ=λB22neffκ2π2+1L2.
(4)

The PbP technique offers unique capabilities in controlling the bandwidth of strong gratings. A reduction in κ could be achieved by reducing the laser pulse energy and therefore Δn. However in our experiment setup it is easier to increase the order m of the grating to reduce the coupling coefficient. Point-by-point gratings have a longitudinal refractive index profile that is closely approximated by a square-wave with a duty cycle that depends on the grating order. Therefore, when creating higher than first-order gratings, we rely on the 1/m relationship that governs the strength of the Fourier resonance orders. When exploiting these higher order, reduced κ gratings, the length of the grating can be easily increased to offset the increased m, and thereby maintain the same κL product (and thus the same overall reflectivity).

We have used the relationship between resonance linewidth and grating order to create narrow resonance gratings of fixed reflectivity. Figure 3
Fig. 3 Bandwidth and coupling coefficient for gratings of orders from 1 to 5. The blue and red trend lines are linear fits of the bandwidth and κ data, the grey trend line is intended as a guide to the eye.
summarizes these results and shows five gratings’ bandwidths as a function of order for gratings written with a fixed pulse energy of 200 nJ. As the order of the gratings was increased from 1 up to 5, the length was proportionately increased from 5 - 25 mm (note that a κL=4 corresponds to a peak grating resonance that is 27 dB deep in transmission or R > 99.8%). We attribute the reduced κL of the first-order grating to an overlapping of the index modulations and therefore a reduced index contrast (this hypothesis is supported by micrography of the grating). We have used this technique for grating bandwidth control to fabricate high reflector and output-coupler cavity mirrors in fibre-laser applications. In a recent report by Williams et al. [19

19. R. J. Williams, N. Jovanovic, G. D. Marshall, and M. J. Withford, “All-optical, actively Q-switched fiber laser,” Opt. Express 18(8), 7714–7723 (2010), http://dx.doi.org/10.1364/OE.18.007714. [CrossRef] [PubMed]

] 8th order gratings were used to create narrow (30 pm FWHM) gratings for applications in a Q-switched fibre laser system.

4. Chirped gratings

The simplest non-uniform grating is the linearly chirped grating structure for which the period varies linearly with position z along the grating length. In order to realize a chirped grating structure the function generator that operated as the laser’s master clock was frequency swept linearly about the 1 kHz base frequency by approximately 0.32 Hz. The start of this sweep was synchronised with the start of the inscription process which was carried out at a fixed velocity. As can be seen from Eq. (1), a chirp of the laser repetition frequency f leads to a chirp in the Bragg wavelength. Note the relatively small change in frequency has no measurable effect on the laser’s performance. The 20 mm long second order grating was designed to have a resonance width increased by 0.25 nm compared to that of an unchirped grating. Figure 4
Fig. 4 Uniform grating (left - green) and chirped grating (right - blue). The widths of the resonances are given at the −3 dB point (equivalent to linear FWHM).
shows the transmission spectrum of the chirped FBG alongside the spectrum of an otherwise identical uniform-period FBG.

The chirped grating structure shown in Fig. 4 has a full-width half-maximum (FWHM) or −3 dB width of 370 pm which is in good agreement (within 10 pm) with the desired 250 pm increase in width over the unchirped grating structure. The maximum reflection strength of the chirped grating is naturally lower than that of the unchirped structure given that the coupling coefficient and length product now has a strong wavelength dependence. (We attribute the irregularity in the chirped grating’s resonance band to small errors in the fabrication process, however one has to keep in mind that in Fig. 4 the logarithmic vertical scale accentuates relatively small perturbations in the reflectivity.) In order to increase the strength of a grating, the length of the grating can be increased. In this instance the practical limit to grating length is set by the mechanical characteristics of the inscription system and using the setup described in Section 2 we can fabricate up to 200 mm long gratings. To increase the magnitude of the chirp the magnitude of the frequency sweep can be increased and a significant chirp of 100 nm would require only a 6.25% chirp of the laser repetition frequency—an amount that can be easily applied without significant perturbation of the laser operating parameters.

5. Amplitude or phase modulated sampled gratings

Periodically modulating the amplitude or phase of a grating’s refractive index perturbations leads to the formation of a sampled or superstructured grating [20

20. B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long Periodic Superstructure Bragg Gratings in Optical Fibers,” Electron. Lett. 30(19), 1620–1622 (1994), http://dx.doi.org/10.1049/el:19941088. [CrossRef]

]. Such gratings exhibit side-band resonances that are approximately equally spaced in frequency about the fundamental (or carrier) resonance. If the period of the sampled modulation in the grating is ΛS, the wavelength spacing of the side-band peaks is given by

ΔλSλB22neffΛS.
(5)

In the simplest case of an applied 50:50 mark-space ratio (MSR) square-wave modulation of the fundamental grating strength, the sideband resonances occur at odd-integer multiples of ΔλS (since the Fourier decomposition of a 50:50 MSR square-wave profile is one with only odd-integer harmonic components). Using the timing electronics shown in Fig. 2 and disabling the delayed trigger line we fabricated a 50 mm long second-order grating with a 100% grating strength modulation, a 50:50 MSR and Λs=1.035  mm (which corresponds to a 100 GHz sideband frequency spacing). The transmission spectrum of the grating is shown in Fig. 5(b)
Fig. 5 Amplitude modulated sampled-grating. (a) Two micrographs of the start and end point of one modulation—the fibre’s core occupies the majority of the image and the grating’s ~1 µm pitch modulations are clearly visible. (b) The transmission spectrum of the grating—the strong central Bragg resonance is accompanied by the sampled-grating sideband resonances.
. The grating exhibits a central Bragg peak with odd harmonic side resonances of decreasing strength extending symmetrically about the centre resonance. On the short wavelength side the sideband resonances are less clear because they overlap the cladding mode resonances [16

16. J. U. Thomas, C. Voigtländer, S. Nolte, A. Tünnermann, N. Jovanovic, G. D. Marshall, M. J. Withford, and M. Steel, “Mode selective fibre-Bragg gratings,” Proc. SPIE 7589, 75890J (2010), http://dx.doi.org/10.1117/12.843805. [CrossRef]

].

One disadvantage of the amplitude modulation approach to sampled grating formation exemplified in Fig. 5 is that the resulting spectrum contains a comb of peaks with irregular frequency spacing and a central triplet with irregular peak heights. In many telecommunication and sensing applications it is often desirable to avoid this and have just two primary peaks or an equally spaced frequency comb. To create a grating with fixed frequency spaced resonances we used a phase modulation approach to completely suppress the central Bragg peak. By creating a delayed laser trigger that represented a π/2 phase-shift at the target Bragg wavelength we fabricated a 50:50 MSR phase modulated sampled-grating. This grating was designed for a dual-wavelength fibre laser application and comprised a 25 mm long second-order structure with a sample period Λs=0.440  mm. This yielded a pair of strong resonance peaks of separation 0.465 THz (or 3.72 nm). The transmission spectrum of the phase modulated device is shown in Fig. 6(b)
Fig. 6 Phase modulated sample grating. (a) Two micrographs of the fibre core in a region containing a phase shift. The whole core is shown on the left and the phase shift region is further magnified on the right. (b) The transmission spectrum of the grating—the fundamental Bragg resonance has been suppressed leaving just the sampled grating sidebands.
and demonstrates that it is possible to null the central Bragg resonance by creating a grating with equal parts that are in and out of phase with one another. The phase shift occurs at the point of the missing modification shown in Fig. 6(a).

6. Single phase-shifted gratings

The ability to precisely control the local phase of the modulations within a grating enables the fabrication of single phase-shifted gratings—a highly useful grating design that exhibits an extremely narrow transmission peak within a central stop-band. The PbP technique allows the post-tuning of a fabricated grating whilst one simultaneously analyses the characteristics of its spectrum. This post-tuning capability is very useful because it is difficult to manufacture a perfect phase-shifted grating in a single fabrication step. These gratings require highly precise control of the modal-phase in the central phase-shifted region. The presence of a grating in a fibre slightly perturbs its modal refractive index and hence writing a desired phase-shift by introducing a controlled length perturbation in the phase of the periods provides only an approximation to the required shift. Hence in most grating manufacturing setups it is necessary to post-tune gratings while simultaneously probing them.

We used the grating-writing laser to modify the core material at the centre of a single phase-shifted grating thereby adjusting the grating’s phase shift. This technique allowed us to precisely tune the pass-band of a phase-shifted grating from the short-wavelength side to the centre of the grating stop band.

In Fig. 7
Fig. 7 Post-manufacture tuning of a single phase-shifted grating. After fabrication (red plot) the grating’s central phase-shift was adjusted in three stages through the orange and green plots to its final value resulting in the blue transmission profile. Each transmission curve is incremented vertically 1 division for clarity.
we show the results of post-tuning a PbP written grating achieved by exposing the central phase-shift region to additional laser pulses (thereby adjusting the refractive index of the core material). The figure shows four transmission profiles of the phase-shifted grating at various stages of the tuning process; note that the position of each plot is incremented by 1 ordinate division to aid clarity. The central phase-shift of the 10 mm long second-order grating was initially written as 0.4π, just less than the required 0.5π shift. While the grating was still in the fabrication setup we post-tuned the central phase. The additional modifications resulting from the laser exposure were not intended to coherently add to the grating but rather to slightly adjust the modal-index in the phase-shift region. Using this post-tuning method it was easy to fabricate gratings and adjust the wavelength of the narrow-pass band region to a desired point within the gratings’ broader stop-band.

7. Amplitude and phase modulated sampled gratings

We exploited these capabilities to create a grating which exhibited a series of 11 narrow-band resonances spaced by 50 GHz (or approximately 0.40 nm at 1545 nm). The grating had an interaction profile that consisted of ten consecutive symmetric sinc envelopes (each containing 6 complete 2π phase revolutions). An example section of the idealized grating modulation profile is shown in Fig. 8
Fig. 8 The ideal coupling coefficient over two envelopes of the sinc profile grating. Positive and negative coupling strength sections are shown in red and blue respectively.
; this figure shows the coupling constant κ(z) for two of the ten sinc envelopes. Because of limitations in the experimental setup used in this study the actual grating coupling profile is rather more complicated than that shown in Fig. 8. This has consequences for the reflection spectrum and these details will be discussed later in section 8.

A 20.7 mm long second-order grating with λB=1550  nm and ΛS=2.07  mmwas fabricated using the methods described above. The reflection spectrum for this grating is shown in Fig. 9(b)
Fig. 9 Combined phase and amplitude modulation grating. (a) A micrograph montage shows the sinc form of the lateral displacement of the grating about the core centre. The horizontal axis of the image is compressed by a factor of four and shows the displacement of the grating from the core centre, across the core/cladding boundary, to the first phase shift at the inversion of the sinc profile. (b) Reflection spectrum from the grating. (c) Modelled reflection spectrum for the same grating presented in (b)—discussed in section 8. (d) Modelled reflection spectrum for idealized PbP grating with no core-centre overshoot—discussed in section 8.
and displays an array of eleven 50 GHz spaced reflection resonances, and a number of low reflectivity ‘satellite’ peaks outside of the main envelope.

In Fig. 9 it can be seen that the reflection strength of the peaks follows an alternating high-low pattern centred about a ‘low’ at the central Bragg resonance. As our gratings are in a weakly-coupled regime, the reflection spectrum can be approximately modelled by obtaining the Fourier transform (FT) of the modulation profile shown in Fig. 8 and this method reproduces the pattern of the 11 main peaks. The FT approach also indicates that the alternating low- and high-peak heights are caused by the odd- and even-harmonic resonances that arise with the truncation of the sinc interaction modulation profile at an integer 2π multiple.

Although it is possible to replicate many of the grating spectrum’s characteristics using a FT, there are some key shortcomings to using this simplistic modelling approach. When performing the grating manufacturing the lateral-position sinc profile was generated using an arbitrary function generator and our simple FT model assumes that the interaction between the grating and the guided mode is linear from the centre of the core (where it is maximum) to the point at the edge of the lateral translation (where the interaction is assumed to be zero). To be more accurate the interaction strength function should include the intensity profile of the core mode and account for the fact that the maximum translation is too close to the core boundary to yield a zero coupling. With these points in mind we opted to develop a more sophisticated grating model based on coupled-mode theory—the results and conclusions of this work are discussed in the following section.

We have shown, that using a PbP system adapted to implement a sinc-coupling modulation technique, it is possible to create gratings with an arbitrary number of resonances. The truncation point of the cyclical sinc modulation profile determines the number of the reflection peaks in the grating spectrum and their degree of alternate peak strength undulation. (Reducing the number No of 2π oscillations of the sinc profile reduces the number of peaks in the resonance spectrum.) The reflection efficiency of gratings written using the interaction-modulation method is naturally lower than those written entirely at the centre of the core—the total integrated coupling between the guided mode and the grating is lower because the majority of the grating is not at the centre of the fibre’s core. However, as is shown in Fig. 9(b), we have fabricated gratings with reflection strengths close to 50%—values eminently suitable for fibre laser and WDM applications. The strength of these gratings can be easily increased by increasing the length of the grating while keeping other parameters constant.

8. Modelling

The actual coupling constant profile κ(x) induced by the PbP writing process is more complicated than the ideal profile shown in Fig. 8. As indicated above, the actual profile is a combined result of the profile of the lateral translation of the grating periods, and the local field intensity of the guided mode.

With the intention of improving our fabrication methods through an improved understanding of our gratings, we developed a grating model based on standard grating coupled-mode equations to obtain the reflection spectrum. The evolution of the forward and backward field amplitudes A± is given by the equations
idA+dz=(δ+σ(z))A++κ(z)AidAdz=(δ+σ(z))A+κ(z)A+,
(6)
where
δ=2πneff(1λ1λB),
(7)
is the frequency detuning of the incoming light from the Bragg wavelength. Using a Gaussian model for the core mode the effective coupling coefficient satisfies
|κ(z)|=κ0e(x(z)w/4)2,
(8)
where w=10.4  μm is the mode field diameter and the transverse displacement of the defects from the centre of the core is given by
x(z)=x1+(x0x1)|sinc2πzΛs/No|.
(9)
The offsets x0 and x1 were obtained from transmission DIC microscope measurements and have an accuracy of approximately 300 nm. The value x1 is the maximum displacement of the modifications from the core. In the ideal case, x0=0, so that the defects at the peak of the sinc function lie at the centre of the core. In practice, our measurements indicate that there is some overshoot so that x0 has a small negative value. The superstructure period Λs and number of oscillations No are known accurately as they are programmed into the manufacturing system. Because the precise morphology and magnitude of the PbP refractive index modulations is not known the coupling constant amplitude κ0 was chosen to fit the measured spectrum. The sign of the coupling coefficient matches that of the function sinc(2πz/(Λs/No)).

Our present experimental method does not preserve a fixed neff over the length of the grating, therefore, as well as the coupling strength, the index modifications also induce a local shift in the detuning of the grating as described in [22

22. J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through Nonuniform Grating Structures,” J. Opt. Soc. Am. A 11(4), 1307–1320 (1994), http://dx.doi.org/10.1364/JOSAA.11.001307. [CrossRef]

]:

σ(z)=2|κ(z)|.
(10)

The sign of the detuning shift is negative to match the sign of the net index change of the modification which is dominated by a central void region [6

6. N. Jovanovic, J. Thomas, R. J. Williams, M. J. Steel, G. D. Marshall, A. Fuerbach, S. Nolte, A. Tünnermann, and M. J. Withford, “Polarization-dependent effects in point-by-point fiber Bragg gratings enable simple, linearly polarized fiber lasers,” Opt. Express 17(8), 6082–6095 (2009), http://dx.doi.org/10.1364/OE.17.006082. [CrossRef] [PubMed]

].

For the grating in Fig. 9(a), we measured displacements of x0=2.25  μm and x1=5.20  μm. Using these parameters, Fig. 10(a)
Fig. 10 (a) Modification displacement function for two periods of the grating in Fig. 9 with x0=2.25  μm and x1=5.20  μm. (b) Coupling coefficient κ(z) (red and blue) and the local detuning σ(z) (green) resulting from (a). (Note that the sign of the detuning is negated for clarity.)
shows two periods of the modification displacement x(z) and Fig. 10(b) shows the resulting coupling coefficient and local detuning. Because the coupling coefficient does not reduce to zero at each phase shift there exist discontinuities in κ(x); these are shown in Fig. 10(b) as vertical black lines connecting the changing sign κ(x) regions. Figure 9(c) shows the calculated reflection spectrum for the grating with coupling and detuning parameters shown in Fig. 10. The calculation accurately predicts the small ancillary satellite resonances in addition to the grating’s main 11 peaks. Using the coupled-mode model we have discovered that the unwanted satellite resonances in the grating reflection spectrum are a result of one source of imperfection in our grating writing system. By reducing the overshoot parameter to x0=0.50  μm the model generates the spectrum in Fig. 9(d), which clearly indicates that the satellite resonances are associated with the dip in the central peak of the coupling coefficient visible in Fig. 10(b).

9. Conclusions

We have demonstrated that the PbP technique for fabricating fibre-Bragg gratings is a versatile and relatively simple method for the inscription of a broad range of complex grating types. The technique has several key advantages over other methods of Bragg grating inscription. They are that the technique can be applied to all waveguide types (because it allows inscription directly into non-photosensitive and doped/active media) and that it is possible to exercise complete autonomy over the location of the refractive index modulations in the waveguide region. This enables unique approaches to controlling the local phase and coupling strength of a grating. In fact it would require only a simple extension to the existing experimental technique to uniquely specify the lateral and longitudinal position of any number of grating periods up to about 65,000 (which is more than three-times the maximum number of periods used in the gratings described here). A further example of where this could be useful is in the realization of anti-symmetric gratings where a forward propagating mode is coupled in reflection into a higher-order (transverse) mode of the fibre. Examples of this type of grating find applications in WDM systems and have been realized both in optical fibres [23

23. M. Åslund, J. Canning, L. Poladian, C. M. de Sterke, and A. Judge, “Antisymmetric grating coupler: experimental results,” Appl. Opt. 42(33), 6578–6583 (2003), http://dx.doi.org/10.1364/AO.42.006578. [CrossRef] [PubMed]

] and planar waveguides [24

24. J. M. Castro, D. F. Geraghty, S. Honkanen, C. M. Greiner, D. Iazikov, and T. W. Mossberg, “Demonstration of mode conversion using anti-symmetric waveguide Bragg gratings,” Opt. Express 13(11), 4180–4184 (2005), http://dx.doi.org/10.1364/OPEX.13.004180. [CrossRef] [PubMed]

]. Using our existing experimental configuration these gratings could be easily manufactured in fibres by positioning the fibre core in alternating lateral positions. Although we do not have the diagnostic equipment to probe such a grating in SMF-28e (which requires a laser source close to the single-mode cut-off wavelength of 1260 nm) we have at least demonstrated the fabrication of such a ‘grating prototype’ whose micrograph is shown in Fig. 11
Fig. 11 Optical micrograph of an anti-symmetric FBG prototype. The grating was fabricated during a single translation process.
below. The alternating lateral position of the grating periods is clearly visible—note that the grating was written in one translation process.

In combination with the methods of phase and coupling control outlined so far the PbP technique offers several other intriguing possibilities for the fabrication of gratings in waveguide platforms besides optical fibres. Point-by-point gratings can be combined with directly-written waveguide structures in glasses [25

25. G. D. Marshall, M. Ams, and M. J. Withford, “Direct laser written waveguide-Bragg gratings in bulk fused silica,” Opt. Lett. 31(18), 2690–2691 (2006), http://dx.doi.org/10.1364/OL.31.002690. [CrossRef] [PubMed]

] and lithographically described polymer waveguides [26

26. A. J. Lee, A. Rahmani, J. M. Dawes, G. D. Marshall, and M. J. Withford, “Point-by-point inscription of narrow-band gratings in polymer ridge waveguides,” Appl. Phys., A Mater. Sci. Process. 90(2), 273–276 (2007), http://dx.doi.org/10.1007/s00339-007-4261-9. [CrossRef]

] in order to create waveguide-Bragg gratings which find applications in sensor and laser source applications. A simple combination of these technologies and those described in [24

24. J. M. Castro, D. F. Geraghty, S. Honkanen, C. M. Greiner, D. Iazikov, and T. W. Mossberg, “Demonstration of mode conversion using anti-symmetric waveguide Bragg gratings,” Opt. Express 13(11), 4180–4184 (2005), http://dx.doi.org/10.1364/OPEX.13.004180. [CrossRef] [PubMed]

] could be used, for example, to fabricate anti-symmetric waveguide-Bragg grating couplers for mode conversion and studies of non-linear coupling in gratings [27

27. S. Ha and A. A. Sukhorukov, “Nonlinear switching and reshaping of slow-light pulses in Bragg-grating couplers,” J. Opt. Soc. Am. B 25(12), C15–C22 (2008), http://dx.doi.org/10.1364/JOSAB.25.000C15. [CrossRef]

]. Another, and as yet unexplored capability of PbP systems, is the use of local phase randomization as a method of grating apodization. We have shown that a phase shift of π/2 can be used to negate the local coupling coefficient of a grating; however the introduction of a controlled amount of random phase error in the range ±π/4 to the position of a single grating modulation could be used to control the strength of coupling. This method of controlling coupling has the advantage that the detuning σ and local neff of the grating remains constant, and systems based on this principle are an area in our ongoing studies.

Acknowledgments

The Authors would like to thank Prof. B. J. Eggleton and Prof. C. M. de Sterke for their helpful discussions. This work was produced with the assistance of the Australian Research Council (ARC) under the Centres of Excellence and Linkage Infrastructure, Equipment and Facilities programs.

References and Links

1.

D. Grobnic, S. J. Mihailov, C. W. Smelser, and H. M. Ding, “Sapphire fiber Bragg grating sensor made using femtosecond laser radiation for ultrahigh temperature applications,” IEEE Photon. Technol. Lett. 16(11), 2505–2507 (2004), http://dx.doi.org/10.1109/Lpt.2004.834920. [CrossRef]

2.

N. Jovanovic, M. Åslund, A. Fuerbach, S. D. Jackson, G. D. Marshall, and M. J. Withford, “Narrow linewidth, 100 W cw Yb3+-doped silica fiber laser with a point-by-point Bragg grating inscribed directly into the active core,” Opt. Lett. 32(19), 2804–2806 (2007), http://dx.doi.org/10.1364/OL.32.002804. [CrossRef] [PubMed]

3.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008), http://dx.doi.org/10.1038/nphoton.2008.47. [CrossRef]

4.

Y. Lai, A. Martinez, I. Khrushchev, and I. Bennion, “Distributed Bragg reflector fiber laser fabricated by femtosecond laser inscription,” Opt. Lett. 31(11), 1672–1674 (2006), http://dx.doi.org/10.1364/OL.31.001672. [CrossRef] [PubMed]

5.

E. Wikszak, J. Thomas, J. Burghoff, B. Ortaç, J. Limpert, S. Nolte, U. Fuchs, and A. Tünnermann, “Erbium fiber laser based on intracore femtosecond-written fiber Bragg grating,” Opt. Lett. 31(16), 2390–2392 (2006), http://dx.doi.org/10.1364/OL.31.002390. [CrossRef] [PubMed]

6.

N. Jovanovic, J. Thomas, R. J. Williams, M. J. Steel, G. D. Marshall, A. Fuerbach, S. Nolte, A. Tünnermann, and M. J. Withford, “Polarization-dependent effects in point-by-point fiber Bragg gratings enable simple, linearly polarized fiber lasers,” Opt. Express 17(8), 6082–6095 (2009), http://dx.doi.org/10.1364/OE.17.006082. [CrossRef] [PubMed]

7.

D. Grobnic, C. W. Smelser, S. J. Mihailov, and R. B. Walker, “Long-term thermal stability tests at 1000°C of silica fibre Bragg gratings made with ultrafast laser radiation,” Meas. Sci. Technol. 17(5), 1009–1013 (2006), http://dx.doi.org/10.1088/0957-0233/17/5/S12. [CrossRef]

8.

A. Martinez, I. Y. Khrushchev, and I. Bennion, “Thermal properties of fibre Bragg gratings inscribed point-by-point by infrared femtosecond laser,” Electron. Lett. 41(4), 176–178 (2005), http://dx.doi.org/10.1049/El:20057898. [CrossRef]

9.

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, “Direct writing of fibre Bragg gratings by femtosecond laser,” Electron. Lett. 40(19), 1170–1172 (2004), http://dx.doi.org/10.1049/El:20046050. [CrossRef]

10.

S. J. Mihailov, C. W. Smelser, P. Lu, R. B. Walker, D. Grobnic, H. M. Ding, G. Henderson, and J. Unruh, “Fiber bragg gratings made with a phase mask and 800-nm femtosecond radiation,” Opt. Lett. 28(12), 995–997 (2003), http://dx.doi.org/10.1364/OL.28.000995. [CrossRef] [PubMed]

11.

D. Grobnic, C. W. Smelser, S. J. Mihailov, R. B. Walker, and P. Lu, “Fiber Bragg gratings with suppressed cladding modes made in SMF-28 with a femtosecond IR laser and a phase mask,” IEEE Photon. Technol. Lett. 16(8), 1864–1866 (2004), http://dx.doi.org/10.1109/LPT.2004.831239. [CrossRef]

12.

J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber Bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys., A Mater. Sci. Process. 86(2), 153–157 (2006), http://dx.doi.org/10.1007/s00339-006-3754-2. [CrossRef]

13.

B. Malo, K. O. Hill, F. Bilodeau, D. C. Johnson, and J. Albert, “Point-by-Point Fabrication of Micro-Bragg Gratings in Photosensitive Fiber Using Single Excimer Pulse Refractive-Index Modification Techniques,” Electron. Lett. 29(18), 1668–1669 (1993), http://dx.doi.org/10.1049/el:19931110. [CrossRef]

14.

A. Martinez, I. Y. Khrushchev, and I. Bennion, “Direct inscription of Bragg gratings in coated fibers by an infrared femtosecond laser,” Opt. Lett. 31(11), 1603–1605 (2006), http://dx.doi.org/10.1364/OL.31.001603. [CrossRef] [PubMed]

15.

M. L. Åslund, N. Nemanja, N. Groothoff, J. Canning, G. D. Marshall, S. D. Jackson, A. Fuerbach, and M. J. Withford, “Optical loss mechanisms in femtosecond laser-written point-by-point fibre Bragg gratings,” Opt. Express 16(18), 14248–14254 (2008), http://dx.doi.org/10.1364/OE.16.014248. [CrossRef] [PubMed]

16.

J. U. Thomas, C. Voigtländer, S. Nolte, A. Tünnermann, N. Jovanovic, G. D. Marshall, M. J. Withford, and M. Steel, “Mode selective fibre-Bragg gratings,” Proc. SPIE 7589, 75890J (2010), http://dx.doi.org/10.1117/12.843805. [CrossRef]

17.

C. W. Smelser, S. J. Mihailov, and D. Grobnic, “Formation of Type I-IR and Type II-IR gratings with an ultrafast IR laser and a phase mask,” Opt. Express 13(14), 5377–5386 (2005), http://dx.doi.org/10.1364/OPEX.13.005377. [CrossRef] [PubMed]

18.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997), http://dx.doi.org/10.1109/50.618322. [CrossRef]

19.

R. J. Williams, N. Jovanovic, G. D. Marshall, and M. J. Withford, “All-optical, actively Q-switched fiber laser,” Opt. Express 18(8), 7714–7723 (2010), http://dx.doi.org/10.1364/OE.18.007714. [CrossRef] [PubMed]

20.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long Periodic Superstructure Bragg Gratings in Optical Fibers,” Electron. Lett. 30(19), 1620–1622 (1994), http://dx.doi.org/10.1049/el:19941088. [CrossRef]

21.

M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, “Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation,” IEEE Photon. Technol. Lett. 10(6), 842–844 (1998), http://dx.doi.org/10.1109/68.681504. [CrossRef]

22.

J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through Nonuniform Grating Structures,” J. Opt. Soc. Am. A 11(4), 1307–1320 (1994), http://dx.doi.org/10.1364/JOSAA.11.001307. [CrossRef]

23.

M. Åslund, J. Canning, L. Poladian, C. M. de Sterke, and A. Judge, “Antisymmetric grating coupler: experimental results,” Appl. Opt. 42(33), 6578–6583 (2003), http://dx.doi.org/10.1364/AO.42.006578. [CrossRef] [PubMed]

24.

J. M. Castro, D. F. Geraghty, S. Honkanen, C. M. Greiner, D. Iazikov, and T. W. Mossberg, “Demonstration of mode conversion using anti-symmetric waveguide Bragg gratings,” Opt. Express 13(11), 4180–4184 (2005), http://dx.doi.org/10.1364/OPEX.13.004180. [CrossRef] [PubMed]

25.

G. D. Marshall, M. Ams, and M. J. Withford, “Direct laser written waveguide-Bragg gratings in bulk fused silica,” Opt. Lett. 31(18), 2690–2691 (2006), http://dx.doi.org/10.1364/OL.31.002690. [CrossRef] [PubMed]

26.

A. J. Lee, A. Rahmani, J. M. Dawes, G. D. Marshall, and M. J. Withford, “Point-by-point inscription of narrow-band gratings in polymer ridge waveguides,” Appl. Phys., A Mater. Sci. Process. 90(2), 273–276 (2007), http://dx.doi.org/10.1007/s00339-007-4261-9. [CrossRef]

27.

S. Ha and A. A. Sukhorukov, “Nonlinear switching and reshaping of slow-light pulses in Bragg-grating couplers,” J. Opt. Soc. Am. B 25(12), C15–C22 (2008), http://dx.doi.org/10.1364/JOSAB.25.000C15. [CrossRef]

OCIS Codes
(320.7130) Ultrafast optics : Ultrafast processes in condensed matter, including semiconductors
(060.3735) Fiber optics and optical communications : Fiber Bragg gratings
(230.7408) Optical devices : Wavelength filtering devices
(060.3510) Fiber optics and optical communications : Lasers, fiber

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: June 15, 2010
Revised Manuscript: August 8, 2010
Manuscript Accepted: August 24, 2010
Published: September 2, 2010

Citation
Graham D. Marshall, Robert J. Williams, Nemanja Jovanovic, M. J. Steel, and Michael J. Withford, "Point-by-point written fiber-Bragg gratings and their application in complex grating designs," Opt. Express 18, 19844-19859 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-19844


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References

  1. D. Grobnic, S. J. Mihailov, C. W. Smelser, and H. M. Ding, “Sapphire fiber Bragg grating sensor made using femtosecond laser radiation for ultrahigh temperature applications,” IEEE Photon. Technol. Lett. 16(11), 2505–2507 (2004), http://dx.doi.org/10.1109/Lpt.2004.834920 . [CrossRef]
  2. N. Jovanovic, M. Åslund, A. Fuerbach, S. D. Jackson, G. D. Marshall, and M. J. Withford, “Narrow linewidth, 100 W cw Yb3+-doped silica fiber laser with a point-by-point Bragg grating inscribed directly into the active core,” Opt. Lett. 32(19), 2804–2806 (2007), http://dx.doi.org/10.1364/OL.32.002804 . [CrossRef] [PubMed]
  3. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2(4), 219–225 (2008), http://dx.doi.org/10.1038/nphoton.2008.47 . [CrossRef]
  4. Y. Lai, A. Martinez, I. Khrushchev, and I. Bennion, “Distributed Bragg reflector fiber laser fabricated by femtosecond laser inscription,” Opt. Lett. 31(11), 1672–1674 (2006), http://dx.doi.org/10.1364/OL.31.001672 . [CrossRef] [PubMed]
  5. E. Wikszak, J. Thomas, J. Burghoff, B. Ortaç, J. Limpert, S. Nolte, U. Fuchs, and A. Tünnermann, “Erbium fiber laser based on intracore femtosecond-written fiber Bragg grating,” Opt. Lett. 31(16), 2390–2392 (2006), http://dx.doi.org/10.1364/OL.31.002390 . [CrossRef] [PubMed]
  6. N. Jovanovic, J. Thomas, R. J. Williams, M. J. Steel, G. D. Marshall, A. Fuerbach, S. Nolte, A. Tünnermann, and M. J. Withford, “Polarization-dependent effects in point-by-point fiber Bragg gratings enable simple, linearly polarized fiber lasers,” Opt. Express 17(8), 6082–6095 (2009), http://dx.doi.org/10.1364/OE.17.006082 . [CrossRef] [PubMed]
  7. D. Grobnic, C. W. Smelser, S. J. Mihailov, and R. B. Walker, “Long-term thermal stability tests at 1000°C of silica fibre Bragg gratings made with ultrafast laser radiation,” Meas. Sci. Technol. 17(5), 1009–1013 (2006), http://dx.doi.org/10.1088/0957-0233/17/5/S12 . [CrossRef]
  8. A. Martinez, I. Y. Khrushchev, and I. Bennion, “Thermal properties of fibre Bragg gratings inscribed point-by-point by infrared femtosecond laser,” Electron. Lett. 41(4), 176–178 (2005), http://dx.doi.org/10.1049/El:20057898 . [CrossRef]
  9. A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, “Direct writing of fibre Bragg gratings by femtosecond laser,” Electron. Lett. 40(19), 1170–1172 (2004), http://dx.doi.org/10.1049/El:20046050 . [CrossRef]
  10. S. J. Mihailov, C. W. Smelser, P. Lu, R. B. Walker, D. Grobnic, H. M. Ding, G. Henderson, and J. Unruh, “Fiber bragg gratings made with a phase mask and 800-nm femtosecond radiation,” Opt. Lett. 28(12), 995–997 (2003), http://dx.doi.org/10.1364/OL.28.000995 . [CrossRef] [PubMed]
  11. D. Grobnic, C. W. Smelser, S. J. Mihailov, R. B. Walker, and P. Lu, “Fiber Bragg gratings with suppressed cladding modes made in SMF-28 with a femtosecond IR laser and a phase mask,” IEEE Photon. Technol. Lett. 16(8), 1864–1866 (2004), http://dx.doi.org/10.1109/LPT.2004.831239 . [CrossRef]
  12. J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber Bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys., A Mater. Sci. Process. 86(2), 153–157 (2006), http://dx.doi.org/10.1007/s00339-006-3754-2 . [CrossRef]
  13. B. Malo, K. O. Hill, F. Bilodeau, D. C. Johnson, and J. Albert, “Point-by-Point Fabrication of Micro-Bragg Gratings in Photosensitive Fiber Using Single Excimer Pulse Refractive-Index Modification Techniques,” Electron. Lett. 29(18), 1668–1669 (1993), http://dx.doi.org/10.1049/el:19931110 . [CrossRef]
  14. A. Martinez, I. Y. Khrushchev, and I. Bennion, “Direct inscription of Bragg gratings in coated fibers by an infrared femtosecond laser,” Opt. Lett. 31(11), 1603–1605 (2006), http://dx.doi.org/10.1364/OL.31.001603 . [CrossRef] [PubMed]
  15. M. L. Åslund, N. Nemanja, N. Groothoff, J. Canning, G. D. Marshall, S. D. Jackson, A. Fuerbach, and M. J. Withford, “Optical loss mechanisms in femtosecond laser-written point-by-point fibre Bragg gratings,” Opt. Express 16(18), 14248–14254 (2008), http://dx.doi.org/10.1364/OE.16.014248 . [CrossRef] [PubMed]
  16. J. U. Thomas, C. Voigtländer, S. Nolte, A. Tünnermann, N. Jovanovic, G. D. Marshall, M. J. Withford, and M. Steel, “Mode selective fibre-Bragg gratings,” Proc. SPIE 7589, 75890J (2010), http://dx.doi.org/10.1117/12.843805 . [CrossRef]
  17. C. W. Smelser, S. J. Mihailov, and D. Grobnic, “Formation of Type I-IR and Type II-IR gratings with an ultrafast IR laser and a phase mask,” Opt. Express 13(14), 5377–5386 (2005), http://dx.doi.org/10.1364/OPEX.13.005377 . [CrossRef] [PubMed]
  18. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997), http://dx.doi.org/10.1109/50.618322 . [CrossRef]
  19. R. J. Williams, N. Jovanovic, G. D. Marshall, and M. J. Withford, “All-optical, actively Q-switched fiber laser,” Opt. Express 18(8), 7714–7723 (2010), http://dx.doi.org/10.1364/OE.18.007714 . [CrossRef] [PubMed]
  20. B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long Periodic Superstructure Bragg Gratings in Optical Fibers,” Electron. Lett. 30(19), 1620–1622 (1994), http://dx.doi.org/10.1049/el:19941088 . [CrossRef]
  21. M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, “Sinc-sampled fiber Bragg gratings for identical multiple wavelength operation,” IEEE Photon. Technol. Lett. 10(6), 842–844 (1998), http://dx.doi.org/10.1109/68.681504 . [CrossRef]
  22. J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through Nonuniform Grating Structures,” J. Opt. Soc. Am. A 11(4), 1307–1320 (1994), http://dx.doi.org/10.1364/JOSAA.11.001307 . [CrossRef]
  23. M. Åslund, J. Canning, L. Poladian, C. M. de Sterke, and A. Judge, “Antisymmetric grating coupler: experimental results,” Appl. Opt. 42(33), 6578–6583 (2003), http://dx.doi.org/10.1364/AO.42.006578 . [CrossRef] [PubMed]
  24. J. M. Castro, D. F. Geraghty, S. Honkanen, C. M. Greiner, D. Iazikov, and T. W. Mossberg, “Demonstration of mode conversion using anti-symmetric waveguide Bragg gratings,” Opt. Express 13(11), 4180–4184 (2005), http://dx.doi.org/10.1364/OPEX.13.004180 . [CrossRef] [PubMed]
  25. G. D. Marshall, M. Ams, and M. J. Withford, “Direct laser written waveguide-Bragg gratings in bulk fused silica,” Opt. Lett. 31(18), 2690–2691 (2006), http://dx.doi.org/10.1364/OL.31.002690 . [CrossRef] [PubMed]
  26. A. J. Lee, A. Rahmani, J. M. Dawes, G. D. Marshall, and M. J. Withford, “Point-by-point inscription of narrow-band gratings in polymer ridge waveguides,” Appl. Phys., A Mater. Sci. Process. 90(2), 273–276 (2007), http://dx.doi.org/10.1007/s00339-007-4261-9 . [CrossRef]
  27. S. Ha and A. A. Sukhorukov, “Nonlinear switching and reshaping of slow-light pulses in Bragg-grating couplers,” J. Opt. Soc. Am. B 25(12), C15–C22 (2008), http://dx.doi.org/10.1364/JOSAB.25.000C15 . [CrossRef]

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