Detuning in apodized point-by-point fiber Bragg gratings : insights into the grating morphology

Point-by-point (PbP) inscription of fiber Bragg gratings using femtosecond laser pulses is a versatile technique that is currently experiencing significant research interest for fiber laser and sensing applications. The recent demonstration of apodized gratings using this technique provides a new avenue of investigation into the nature of the refractive index perturbation induced by the PbP modifications, as apodized gratings are sensitive to variation in the average background index along the grating. In this work we compare experimental results for Gaussianand sinc-apodized PbP gratings to a coupled-mode theory model, demonstrating that the refractive index perturbation induced by the PbP modifications has a negative contribution to the average background index which is small, despite the presence of strong reflective coupling. By employing Fourier analysis to a simplified model of an individual modification, we show that the presence of a densified shell around a central void can produce strong reflective coupling with near-zero change in the average background index. This result has important implications for the experimental implementation of apodized PbP gratings, which are of interest for a range of fiber laser and fiber sensing technologies. © 2013 Optical Society of America OCIS codes: (060.3735) Fiber Bragg gratings; (060.3510) Lasers, fiber; (230.7408) Wavelength filtering devices; (060.2370) Fiber optics sensors; (320.7130) Ultrafast processes in condensed matter, including semiconductors; (140.3390) Laser materials processing. References and links 1. A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, “Direct writing of fibre Bragg gratings by femtosecond laser,” Electron. Lett. 40, 1170–1172 (2004). 2. J. Thomas, C. Voigtländer, R. G. Becker, D. Richter, A. Tünnermann, and S. Nolte, “Femtosecond pulse written fiber gratings: A new avenue to integrated fiber technology,” Laser Photonics Rev. 6, 709–723 (2012). 3. N. Jovanovic, A. Fuerbach, G. D. Marshall, M. J. Withford, and S. D. Jackson, “Stable high-power continuouswave Yb3+-doped silica fiber laser utilizing a point-by-point inscribed fiber Bragg grating,” Opt. Lett. 32, 1486– 1488 (2007). 4. A. Stefani, M. Stecher, G. E. Town, and O. Bang, “Direct writing of fiber Bragg grating in microstructured polymer optical fiber,” IEEE Photonics Technol. Lett. 24, 1148–1150 (2012). 5. R. J. Williams, N. Jovanovic, G. D. Marshall, G. N. Smith, M. J. Steel, and M. J. Withford, “Optimizing the net reflectivity of point-by-point fiber Bragg gratings: The role of scattering loss,” Opt. Express 20, 13451–13456 (2012). #195737 $15.00 USD Received 13 Aug 2013; revised 16 Oct 2013; accepted 19 Oct 2013; published 30 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.026854 | OPTICS EXPRESS 26854 6. J. Thomas, N. Jovanovic, R. G. Becker, G. D. Marshall, M. J. Withford, A. Tünnermann, S. Nolte, and M. J. Steel, “Cladding mode coupling in highly localized fiber Bragg gratings: Modal properties and transmission spectra,” Opt. Express 19, 325–341 (2011). 7. R. Goto, R. J. Williams, N. Jovanovic, G. D. Marshall, M. J. Withford, and S. D. Jackson, “Linearly polarized fiber laser using a point-by-point Bragg grating in a single-polarization photonic bandgap fiber,” Opt. Lett. 36, 1872–1874 (2011). 8. J. Burgmeier, W. Schippers, N. Emde, P. Funken, and W. Schade, “Femtosecond laser-inscribed fiber Bragg gratings for strain monitoring in power cables of offshore wind turbines,” Appl. Opt. 50, 1868–1872 (2011). 9. G. D. Marshall, R. J. Williams, N. Jovanovic, M. J. Steel, and M. J. Withford, “Point-by-point written fiber-Bragg gratings and their application in complex grating designs,” Opt. Express 18, 19844–19859 (2010). 10. T. Geernaert, K. Kalli, C. Koutsides, M. Komodromos, T. Nasilowski, W. Urbanczyk, J. Wojcik, F. Berghmans, and H. Thienpont, “Point-by-point fiber Bragg grating inscription in free-standing step-index and photonic crystal fibers using near-IR femtosecond laser,” Opt. Lett. 35, 1647–1649 (2010). 11. C. Koutsides, K. Kalli, D. J. Webb, and L. Zhang, “Characterizing femtosecond laser inscribed Bragg grating spectra,” Opt. Express 19, 342–352 (2011). 12. R. J. Williams, N. Jovanovic, G. D. Marshall, and M. J. Withford, “All-optical, actively Q-switched fiber laser,” Opt. Express 18, 7714–7723 (2010). 13. B. Malo, K. O. Hill, F. Bilodeau, D. C. Johnson, and J. Albert, “Point-by-point fabrication of micro-Bragg gratings in photosensitive fibre using single excimer pulse refractive index modification techniques,” Electron. Lett. 29, 1668–1669 (1993). 14. E. Wikszak, J. Burghoff, M. Will, S. Nolte, A. Tünnermann, and T. Gabler, “Recording of fiber Bragg gratings with femtosecond pulses using a “point by point” technique,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2004), p. CThM7. 15. J. U. Thomas, N. Jovanovic, R. G. Krämer, G. D. Marshall, M. J. Withford, A. Tünnermann, S. Nolte, and M. J. Steel, “Cladding mode coupling in highly localized fiber Bragg gratings II: Complete vectorial analysis,” Opt. Express 20, 21434–21449 (2012). 16. S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effectivearea, higher-order mode fibers: A new strategy for high-power lasers,” Laser Photonics Rev. 2, 429–448 (2008). 17. J. Albert, L. Y. Shao, and C. Caucheteur, “Tilted fiber Bragg gratings sensors,” Laser Photonics Rev. 7, 83–108 (2012). 18. L. Jin, Z. Wang, Q. Fang, Y. Liu, B. Liu, G. Kai, and X. Dong, “Spectral characteristics and bend response of Bragg gratings inscribed in all-solid bandgap fibers,” Opt. Express 15, 15555–15565 (2007). 19. R. J. Williams, C. Voigtländer, G. D. Marshall, A. Tünnermann, S. Nolte, M. J. Steel, and M. J. Withford, “Pointby-point inscription of apodized fiber Bragg gratings,” Opt. Lett. 36, 2988–2990 (2011). 20. A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, “Photoinduced modifications in fiber gratings inscribed directly by infrared femtosecond irradiation,” IEEE Photonics Technol. Lett. 18, 2266–2268 (2006). 21. 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, 6082–6095 (2009). 22. Y. Lai, K. Zhou, K. Sugden, and I. Bennion, “Point-by-point inscription of first-order fiber Bragg grating for C-band applications,” Opt. Express 15, 18318–18325 (2007). 23. A. Martinez, I. Y. Khrushchev, and I. Bennion, “Thermal properties of fibre Bragg gratings inscribed point-bypoint by infrared femtosecond laser,” Electron. Lett. 41, 176–178 (2005). 24. V. Mizrahi and J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993). 25. J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994). 26. M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, and S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31, 1488–1490 (1995). 27. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997). 28. R. Feced and M. N. Zervas, “Effects of random phase and amplitude errors in optical fiber Bragg gratings,” J. Lightwave Technol. 18, 90–101 (2000). 29. C. Voigtländer, P. Zeil, J. Thomas, M. Ams, R. J. Williams, M. J. Withford, A. Tünnermann, and S. Nolte, “Fs laser induced apodised Bragg waveguides in fused silica,” Proc. SPIE 7925, 79250Y (2011). 30. S. Juodkazis, H. Misawa, T. Hashimoto, E. G. Gamaly, and B. Luther-Davies, “Laser-induced microexplosion confined in a bulk of silica: Formation of nanovoids,” Appl. Phys. Lett. 88, 201909 (2006). 31. M. L. Åslund, N. Jovanovic, 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, 14248–14254 (2008). 32. T. Hashimoto, S. Juodkazis, and H. Misawa, “Void recording in silica,” Appl. Phys. A 83, 337–340 (2006). 33. E. N. Glezer and E. Mazur, “Ultrafast-laser driven micro-explosions in transparent materials,” Appl. Phys. Lett. #195737 $15.00 USD Received 13 Aug 2013; revised 16 Oct 2013; accepted 19 Oct 2013; published 30 Oct 2013 (C) 2013 OSA 4 November 2013 | Vol. 21, No. 22 | DOI:10.1364/OE.21.026854 | OPTICS EXPRESS 26855 71, 882–884 (1997). 34. E. Toratani, M. Kamata, and M. Obara, “Self-fabrication of void array in fused silica by femtosecond laser processing,” Appl. Phys. Lett. 87, 171103 (2005). 35. S. Kanehira, J. Si, J. Qiu, K. Fujita, and K. Hirao, “Periodic nanovoid structures via femtosecond laser irradiation,” Nano Lett. 5, 1591–1595 (2005). 36. X. Wang, F. Chen, Q. Yang, H. Liu, H. Bian, J. Si, and X. Hou, “Fabrication of quasi-periodic micro-voids in fused silica by single femtosecond laser pulse,” Appl. Phys. A 102, 39–44 (2011).


Introduction
Point-by-point (PbP) inscription of fiber Bragg gratings (FBGs) using femtosecond laser pulses [1] is a flexible technique that is experiencing both intense research interest and broad application, particularly for fiber lasers and fiber sensing [2][3][4][5][6][7][8][9][10][11][12].This technique benefits from the freedom to determine the Bragg period without a phase-mask [1,13,14], as well as the ability to localize modifications within or outside of the core with sub-micron precision [9].The latter facilitates tremendous freedom over the coupling to cladding mode resonances [6,15], which offers exciting potential for applications in sensing and mode-conversion [16,17].These localized modifications have also been used to inscribe a grating in the silica core of a photonicbandgap fiber [7], thereby eliminating detrimental modification to the surrounding photosensitive regions [18]; and have enabled the development of non-uniform PbP gratings, including phase-shifted, superstructure, linearly-chirped and apodized PbP gratings [9,19].
PbP gratings consist of unique refractive index modifications: each single-pulse modification comprising a micro-void encased in an elliptical, densified shell [20,21].The morphology and refractive index composition of these structures is an important characteristic as it affects both the physical and spectral qualities of these gratings, such as high-temperature stability, coupling to cladding-modes, birefringence and polarization-dependent grating strength [6,15,[21][22][23].Furthermore, the refractive index profile can strongly affect the spectral response of apodized gratings.In the recent demonstration of apodized PbP gratings it was noted that they exhibit effective sidelobe suppression on both sides of the main reflection peak [19].However, for an apodization technique that only tailors the magnitude of the coupling coefficient κ(z), it is expected that strong sidelobes would still be present on one side of the main reflection peak [24] (either the short-wavelength side in the case of a positive refractive index perturbation, or the long-wavelength side in the case of a negative perturbation).This is due to the induced variation in the average background index n(z) (averaged over a single grating period), which results in a detuning of the Bragg resonance that varies along the length of the grating [25].Therefore the unexpected result of symmetric sidelobe suppression in our apodized PbP gratings indicated that the relationship between the magnitude of the index perturbation ∆n(z) and the average background index n(z) in PbP gratings may be very different to that of fiber Bragg gratings inscribed with more conventional techniques.
In this work we exploit the spectral properties of apodized PbP gratings to reveal new insights into the refractive index modifications which constitute femtosecond laser-inscribed PbP gratings.In particular, the sensitivity in the spectral response of apodized gratings to variations in the average background index along the grating provides a means to study the net refractive index contribution from the femtosecond laser-inscribed PbP modifications.By comparing experimental results for Gaussian-and sinc-apodized gratings to a model based on the grating coupled-mode equations, we show that whilst the modifications induce a refractive index perturbation ∆n(z) that provides strong retro-reflective coupling (i.e. a strong coupling coefficient κ), the net contribution to the average background index n(z) is small and negative.This is due to the heterogeneous morphology of the PbP modifications, which consist of a micro-void that has a lower refractive index than the fiber core, encased in a densified shell that has a higher refractive index than the unmodified core.We also present scanning electron micrographs of cross-sectioned PbP gratings, revealing the size and structure of the voids within the PbP modifications.Through Fourier analysis of a simplified model of an individual PbP modification we show that such a modification can produce a strong coupling coefficient accompanied by a near-zero change in the average background index n(z).This characteristic of femtosecond laser-inscribed PbP gratings provides inherent simplicity in the realization of apodized PbP gratings and is particularly fortuitous as tailoring or post-tuning the average background index profile using PbP inscription would be highly challenging or involve additional processing steps.Furthermore, these results provide further confirmation of the net negative refractive index contribution of femtosecond laser-inscribed PbP modifications, first inferred by other means in [21].

The effect of detuning on apodized gratings
It was discovered early on in the development of fiber Bragg gratings that changes in the average background index n(z) along the grating causes potentially undesirable effects in the spectral response of apodized gratings [24,25].Due to the Bragg condition, changes in the average background index result in a proportional shift in the local Bragg wavelength.In the case of apodized gratings in which the induced index perturbation is entirely positive (or negative), there is a proportionate position-dependent shift in the local average background index, as illustrated in Fig. 1(a).In the case of strong gratings this can produce a Fabry-Pérot cavity between the two ends of the grating which, due to the detuning, have a local Bragg resonance that does not overlap with the stop-band of the main portion of the grating (see Fig. 1(b)) [25].This Fabry-Pérot effect produces sharp transmission notches on one side of the main stop-band (see Fig. 1(c)) [24].This detuning effect and the associated sidelobes can be mitigated by ensuring that the average background index is constant along the length of the grating.This is commonly achieved in conventional UV inscription of apodized gratings using the phase-mask dithering technique, where there is a varying perturbation amplitude ∆n(z) due to the 'washing out' of the grating periods by the dithering of the phase mask; yet n(z) remains constant as there is a constant dose of UV irradiation along the grating [26].

The coupled-mode-theory model
In order to model the spectral response of apodized PbP gratings we use the standard grating coupled-mode equations, with the addition of a loss term to account for the broadband scattering loss observed in these gratings [5].Thus the evolution of the forward-and backwardpropagating field amplitudes, A + and A − , along the fiber axis z can be described as: where is the wavenumber detuning of the incident light from the Bragg wavelength λ B , and is the wavenumber detuning of the local Bragg resonance due to variation in the average background index (n 0 is the effective index of the guided mode in the unperturbed fiber).Due to the small size of the PbP modifications compared to the width of the core-guided mode, the coupling coefficient κ(z) has an approximately Gaussian dependence upon the offset x(z) of the modifications from the center of the fiber core [9,19], such that where w = 10.4 µm is the 1/e 2 width of the core mode in SMF-28e fiber.The broadband scattering loss induced by the modifications is proportional to their overlap with the field profile of the core mode, therefore we express the loss term as where α 0 is a constant.Thus the transmission loss due to the broadband scattering (typically measured outside the grating stop-band) satisfies where L is the length of the grating (this being a more general expression of Eq. ( 1) from [5]).
The detuning σ (z) due to the local change in the average background index induced by the grating modifications can be similarly expressed as (σ 0 is a constant), due to its proportionality to the overlap of the modifications with the field profile of the core mode.It should be noted that in the case of conventional UV laser-inscribed gratings where the induced refractive index modification is positive, uniform in cross-section, and extends across the core of the fiber, σ (z) = 2|κ(z)|, or equivalently σ 0 = 2|κ 0 | [27].Recalling from grating coupled-mode theory that the stop-band of a grating is given by the range |δ | < κ, we see that having σ = 2|κ| facilitates the formation of a Fabry-Pérot cavity as illustrated in Fig. 1(b).Therefore, in order to model our apodized gratings we need to provide the coupling strength, detuning and scattering loss constants κ 0 , σ 0 , and α 0 , and the function x(z) that defines the offset of the modifications from the centre of the fiber core.x(z) is determined in our experimental setup by the voltage waveform that we use to drive the piezo-electric stage that controls the position of the fiber with respect to the focal point of the laser [9,19].The peak coupling strength κ 0 of a grating can be determined by measuring the transmission extinction of the Bragg resonance.Similarly, a measurement of the out-of-band transmission loss yields the magnitude of the scattering loss coefficient (see Eq. ( 6)).We estimate the detuning coefficient σ 0 by comparison of the modelled and experimental spectra.

Point-by-point inscription of apodized gratings
Gaussian-apodized and sinc-apodized PbP gratings were fabricated in Corning SMF-28e optical fiber with target wavelengths λ B in the range 1520-1570 nm using focussed 800 nm femtosecond laser pulses and a fiber-guiding system with sub-micrometer transverse control.The grating inscription technique is described in detail in [9].The gratings were characterized in transmission and reflection using a high-resolution (3 pm) swept wavelength system (JDSU 15100) in conjunction with a C-band fiber circulator.

Gaussian-apodized PbP gratings
Gaussian-apodized gratings were fabricated by translating the fiber in a straight line within the focal plane of the objective, such that the modifications of the grating trace a line across the core of the fiber, starting and ending a few microns outside the core and crossing through the center of the core in the middle of the grating.This linear variation in the radial offset of the grating modifications maps onto the Gaussian profile of the core mode, giving rise to a Gaussian-apodization profile (as illustrated in Fig. 2) [19].

Sinc-apodized PbP gratings
As described in [19], the translation function necessary for producing a sinc-apodization profile using this PbP apodization method is derived by substituting the desired sinc apodization profile into Eq.( 4) and solving for x(z), which yields the translation function where N 0 is the number of 2π oscillations in the truncated sinc wave.This translation function must be truncated to some maximum value x max that guarantees κ(x max ) κ 0 , which was in this case chosen to be 6 µm (resulting in κ(x max ) = 0.005 × κ 0 ).At each of these maxima in x(z)-which correspond to a zero-crossing in the desired sinc apodization function-a π phaseshift is introduced to change the sign of κ(z).This apodization scheme is illustrated in Fig. 3.The combination of this design function and the measurement of the extremities of the radial offset positions in a grating enables determination of the actual translation function in a grating, which can then be used for modelling the grating's spectral response.

Detuning in Gaussian-apodized PbP gratings
Figure 4 shows three different sets of modelled grating spectra (dashed curves), each compared against the measured transmission and reflection spectrum of a Gaussian-apodized PbP grating (solid curves).The grating is 10 mm long, was inscribed with 230 nJ pulses, and has a second order Bragg resonance at 1522.9 nm with a minimum transmission of approximately −29 dB.Each of the modelled spectra use the following grating parameters: L = 10 mm, κ 0 = 0.8 mm By comparing the graphs in Fig. 4, we can see that the closest agreement between experiment and the model occurs when we set σ (z) = 0 (Fig. 4(c)).In the case σ (z) = −2|κ(z)| (Fig. 4(a))-which would apply for a uniform, negative refractive index modification across the fiber core-there is strong asymmetry in the modelled spectra, both in the shape of the main peak and the presence of sidelobes on the long-wavelength side of this peak (the anticipated in-gap Fabry-Pérot resonances).The case σ (z) = −0.5|κ(z)|shown in Fig. 4(b), presents almost comparable agreement between experiment and model as that observed in Fig. 4(c).Additionally, we see that the slight asymmetry in the experimental spectra-particularly the difference in slope of the band-edges as observed in reflection-corresponds to the asymmetry in the modelled spectra for σ (z) = −0.5|κ(z)|:namely the long-wavelength band-edge in reflection is steeper than the short-wavelength band-edge.This indicates that σ (z) ≤ 0 for this grating (otherwise the asymmetry would be reversed about the Bragg wavelength).Therefore we can reasonably estimate that for this grating, −0.5|κ(z)| ≤ σ (z) ≤ 0.
We note that the measured spectra in Fig. 4 feature irregular sidelobes that are not present in the modelled spectra.Although the cause of these sidelobes is not well understood, random phase errors are known to produce irregular sidelobes in apodized fiber Bragg gratings [28,29].Therefore we suspect these are due to random phase errors in our PbP gratings, which are observed in micrographs of the gratings (see Fig. 5) and are due to vibrations in the fiber as it is drawn through the ferrule.Size variations in the modifications (due to pulse-energy fluctuations) may also contribute to this effect via local variations in the average background index.

Sinc-apodized PbP gratings
Similarly to Fig. 4, Fig. 6 compares the measured transmission and reflection spectra of a sincapodized grating with modelled spectra for three different values for σ (z).The grating is 60 mm long, was inscribed with 220 nJ pulses and has a second order resonance at 1541.1 nm.Subsequent microscopy of the grating showed that the translation function stretched from the center of the core to a maximum offset of 8.5 µm.This is accounted for in the model of the grating.The number of 2π oscillations in the truncated sinc wave N 0 was 7.8.The modelled spectra of the grating include a coupling strength amplitude κ 0 = 0.44 mm −1 (which in the case of a conventional grating perturbation that is sinusoidal in z and uniform across the core of a fiber would correspond to an index perturbation ∆n = 2.54 × 10 −4 ).In the same way as in Fig. 4, the strength of the detuning parameter is varied between the modelled spectra in Fig. 6 Once again, comparing the three cases in Fig. 6, we observe that the closest agreement between experiment and model occurs for the cases σ (z) = 0 and σ (z) = −0.5|κ(z)| in the model (Figs.6(b) and 6(c)).Fig. 6(a) shows strong asymmetry in the main reflection peak which does not correspond to the experimental spectra.There is a small degree of asymmetry in the experimental spectra which is also present, but to a more marked extent in the modelled spectra of Fig. 6(b).Therefore these results indicate that again for this grating, −0.5|κ(z)| ≤ σ (z) ≤ 0, which is of smaller magnitude and opposite sign to the case of conventional UV laser-inscribed gratings where the induced refractive index modification is positive and uniform across the core of the fiber, for which σ (z) = 2|κ(z)|.There remains some difference in shape and bandwidth between the experiment and modelled spectrum in Fig. 6(c), where the experimental spectrum exhibits a broader resonance with steeper band-edges and a more sharp-cornered flat-top profile.The source of these discrepancies is not yet known; however, there may be contributions from error in the experimental implementation of the translation function x(z), due to the inertia of the positioning stages and the velocity at which it translates during fabrication.

Defining the model
As mentioned above, in conventional UV Bragg gratings with no dithering the local detuning and coupling satisfy σ (z) = 2|κ(z)| [25].We have demonstrated that this is not the case for our PbP gratings.Here we identify the source of this different behavior as associated with the three-dimensional morphology of the PbP modifications.
The relative permittivity distribution of the fiber and grating can be written in the general form ε(x, y, z) = ε bg (x, y) + δ ε(x, y, z), where ε bg (x, y) = n 0 (x, y) 2 is the background waveguide cross section and δ ε(x, y, z) is the change in permittivity associated with the grating structure.This function may be expanded in a general Fourier series as where we have introduced a slow longitudinal coordinate Z to describe the variation associated with the apodization on the scale of many periods.In conventional holographic gratings, we simply have ε(x, y, z) = n 0 (x, y) + ∆n(x, y) 1 + cos 2πz where we have dropped higher spatial frequency components that are not phase-matched for a first-order grating.We can then identify δ ε 0 = δ ε 1 = n 0 ∆n(x, y).However in a PbP grating consisting of localized voids and densified regions, the expansion coefficients in Eq. ( 10), found from ε(x, y, z, Z)e −i2π jz/Λ dz, (12) have no particular relation to each other.By standard techniques of coupled-mode theory one can show that the governing coupledmode equations, Eq. ( 1), can be written in the more general form where the coefficients γ i describe the interaction with the grating and for longitudinally symmetric modifications are defined by where f (x, y) is the field profile of the mode incident on the grating.In terms of these parameters, the local detuning is and the coupling strength for the mth order grating is For the standard grating we thus recover σ (z) = 2|κ(z)|, but for more general index modifications such as with PbP gratings, this is not the case.It is apparent for instance that the γ m coefficient is likely to be largest when the width of the modification satisfies w ≈ Λ/(2m).

Measurement of Type II-IR voids
In order to attain accurate data from the model we would need to measure the exact threedimensional refractive index profile of the PbP modifications, including both the densified shell and the microvoid; however this kind of measurement, requiring high accuracy and sub-micron resolution within a buried, composite structure is probably not achievable with any current refractive index profiling technique.Furthermore, we anticipate significant variation in the modified features from day-to-day and even from pulse-to-pulse, as the laser material interaction is highly nonlinear and thus highly sensitive to pulse peak power variations, the phase profile of the pulse, etc.Therefore, we use this model only to illustrate the effects of the complex morphology of these features on κ and σ , using a very simplified representation of the actual PbP modifications.However, previous investigations into the morphology of PbP grating modifications have been limited to optical microscope observations [20] and information gained from analysis of the gratings spectral features (such as net negative contribution to the average background index) [21].In particular, to our knowledge, an accurate measurement of the size and shape of the void structures in PbP gratings has not been reported (i.e. with resolution 1 µm).Therefore in order to provide parameters for a model that accounts for the modification morphology, we first measured the size and shape of the voids within PbP gratings using scanning electron microscopy (SEM).
In order to reveal the modifications from within the fiber we cross-sectioned PbP gratings using an argon-ion cross-sectional polisher (JEOL IB-09010CP).This method was chosen to avoid contaminating the voids or preferentially etching/polishing the voids or the densified shell, as is the case, for example, with wet etching [30].Gratings were polished at an angle so as to section a series of voids at a time, at different points within each void.Images of a cross-sectioned fiber are shown in Fig. 7.
Cross-sectioned PbP grating samples were then imaged, without pre-coating, using a highresolution field-emission SEM (JEOL JSM-7800F).Rather than observing a single, elliptical micro-void per femtosecond pulse modification, of dimensions approximately 0.5-2 µm, as previously reported in [6,20,31]; we observe a chain of spherical voids, of dimensions <400 nm, for each femtosecond laser pulse.The chain of voids are aligned in the propagation direction of the inscribing femtosecond laser.SEM micrographs for three different PbP gratings are shown in Fig. 8.Note that the rotational alignment of the grating planes to the cross-sectioning plane is not perfectly consistent, and so in some images the grating planes appear to have a diagonal slant in their orientation.This also means that only one set of voids are cross-sectioned through their center; whereas the voids above and below these are crosssectioned away from center and therefore appear smaller.The grating in Fig. 8 The formation of chains of multiple voids by a single femtosecond pulse has not commonly been observed in bulk silica glass under similar writing conditions (e.g.focussing N.A. and pulse energy), where single void formation is more prevalent [32,33], or multiple-void forma-tion typically requires multiple-pulse irradiation [34,35].multiple-void formation by a single femtosecond pulse has been observed for higher pulse energies with similar focussing conditions (0.8 N.A., 30 fs pulse duration, 3-5 µJ pulse energy), as reported in [36].The curved refracting interfaces between the inscription lens and the fiber core in PbP grating inscription (such as the core/cladding boundary), which are not present in bulk-glass processing, may be contributing to the fact that we observe elongated chains of several voids with single pulse inscription at low pulse energy; however this requires further investigation.To date, measurement of PbP modifications has been mainly limited to optical microscopy and thus has been limited by the optical resolution of the microscope [6,20,21,31]; whereas the ion-beam crosssectioning technique used in this work has enabled direct, high-resolution measurement using a SEM, without risk of differential etching or smearing effects which can occur with mechanical polishing or wet-etching [30].

Modelling results
In order to implement the model and illustrate the effect of the composite PbP modifications on κ and σ , we consider the void chains as a single ellipsoidal void of width w v and height h v to approximate the total void volume in a single grating period, using a range of values according to our SEM observations.Due to the larger size of the densified shell and the fact that they are not visible under SEM observation, we rely on DIC micrographs (such as Fig. 5) for indicative sizes of these regions (width w s and height h s ).We consider the void to have refractive index 1 and we estimate the refractive index of the shell to be uniform and determined simply by densification due to the material evacuated from the void.Clearly this model is highly idealized; however, as mentioned above, an accurate measurement of the refractive index profile of the shell would be extremely challenging, and this model still serves to illustrate the effect of the complex morphology on κ and σ .Figure 9 shows calculations of σ (blue), κ (red) and the ratio σ /κ (black) as a function of the void width w v for two configurations.Figure 9(a) models the case of a spherical void (void ellipticity η v = h v /w v = 1) encased in a shell of height 4 µm, so that the values near w v = 80 µm may be considered representative of the smaller defects in Fig. 8(a).Figure 9(b) considers elliptical voids where the vertical axis is three times larger (η v = 3), as a qualitative description of the chains of three voids observed in Figs.8(b) and 8(c).In this case the shell height h s = 8 µm, which is representative of modifications inscribed at higher pulse energies.While the precise values depend on the particular choices made and these examples are to be regarded as illustrative only, it is clear that the different averaging of the detuning and coupling constant lead to values of the ratio σ /κ which are for the most part negative and in the range −|κ| < σ < 0, consistent with our fits of the measured reflection spectra.

Conclusions
We have shown that the unique morphology of the modifications in femtosecond laser-inscribed PbP gratings, which consist of a micro-void encased in a densified shell, gives rise to a net change in the average background index that is small and negative, whilst at the same time producing strong reflective coupling.Our analysis of the coupling properties of a simplified PbP modification illustrate that a single modification consisting of decreasing and increasing index perturbations (i.e.−∆n and +∆n regions, corresponding to a void and a densified shell, respectively) can simultaneously give rise to a strong coupling coefficient whilst inducing near-zero net change in the average background index (and thus near-zero detuning).Our demonstration of the net negative change in the average background index in femtosecond laser-inscribed PbP gratings confirms the result inferred in [21] which was based on polarization-dependent spectral measurements.This result is also in agreement with the observations of positive and negative refractive index modifications within PbP gratings reported in [20].
We have also presented scanning electron micrographs of the void structures within PbP gratings, showing that each femtosecond pulse creates a chain of spherical voids and that these voids vary in size from <100 nm up to approximately 400 nm (within the specified pulse energy range which is typical for PbP grating inscription).
These results shed new light on the refractive index composition and morphology of these PbP structures and reveal useful insights into the effect of these modifications on PbP fiber grating spectra.They also demonstrate that our PbP apodization technique, which only considers the coupling strength profile in the design, is capable of producing apodized gratings that have negligible detuning effects because of the inherent characteristics of femtosecond PbP gratings.This is advantageous for PbP inscription as it is not compatible with conventional techniques for tailoring or post-tuning the average background index in a grating.

Fig. 1 .
Fig. 1.(a) Illustration of the refractive index profile of a Gaussian-apodized grating with exaggerated period.(b) Band-diagram representation of such a Gaussian-apodized grating showing a Fabry-Pérot cavity on the short-wavelength side of the main stop-band, and the relative magnitudes of the detuning and coupling constants, σ and κ.(c) Modelled reflection spectra of a Gaussian-apodized grating with positive index modifications, exhibiting strong transmission notches on the short-wavelength side of the main reflection peak.
Figure4shows three different sets of modelled grating spectra (dashed curves), each compared against the measured transmission and reflection spectrum of a Gaussian-apodized PbP grating (solid curves).The grating is 10 mm long, was inscribed with 230 nJ pulses, and has a second order Bragg resonance at 1522.9 nm with a minimum transmission of approximately −29 dB.Each of the modelled spectra use the following grating parameters: L = 10 mm, κ 0 = 0.8 mm −1 , x(0) = 5 µm and x(L) = 4 µm.The strength of the detuning parameter is varied between these modelled spectra, with values σ (z) = −2|κ(z)|, σ (z) = −0.5|κ(z)|and σ (z) = 0 corresponding to Figs. 4(a), 4(b) and 4(c), respectively.By comparing the graphs in Fig.4, we can see that the closest agreement between experiment and the model occurs when we set σ (z) = 0 (Fig.4(c)).In the case σ (z) = −2|κ(z)| (Fig.4(a))-which would apply for a uniform, negative refractive index modification across the fiber core-there is strong asymmetry in the modelled spectra, both in the shape of the main peak and the presence of sidelobes on the long-wavelength side of this peak (the anticipated in-gap Fabry-Pérot resonances).The case σ (z) = −0.5|κ(z)|shown in Fig.4(b), presents almost comparable agreement between experiment and model as that observed in Fig.4(c).Additionally, we see that the slight asymmetry in the experimental spectra-particularly the difference in slope of the band-edges as observed in reflection-corresponds to the asymmetry in the modelled spectra for σ (z) = −0.5|κ(z)|:namely the long-wavelength band-edge in reflection is steeper than the short-wavelength band-edge.This indicates that σ (z) ≤ 0 for this grating (otherwise the asymmetry would be reversed about the Bragg wavelength).Therefore we can reasonably estimate that for this grating, −0.5|κ(z)| ≤ σ (z) ≤ 0.We note that the measured spectra in Fig.4feature irregular sidelobes that are not present in the modelled spectra.Although the cause of these sidelobes is not well understood, random phase errors are known to produce irregular sidelobes in apodized fiber Bragg gratings[28,29].Therefore we suspect these are due to random phase errors in our PbP gratings, which are observed in micrographs of the gratings (see Fig.5) and are due to vibrations in the fiber as it is drawn through the ferrule.Size variations in the modifications (due to pulse-energy fluctuations) may also contribute to this effect via local variations in the average background index.

Fig. 5 .
Fig. 5. Differential-interference-contrast (DIC) micrographs of the extremities of the Gaussian-apodized grating.The top images are viewed from the direction of the inscribing beam; the bottom images are viewed from the orthogonal direction.

FiberFig. 7 .Fig. 8 .
Fig. 7. Optical micrographs of a cross-sectioned PbP FBG: (a) side view; (b) top view.The fiber can be seen protruding at an angle from between two glass coverslips (see (a)).The polished end-face of the fiber features in the centre of image (b), and is elliptical due to the angle of the fiber with respect to the polishing axis.

Fig. 9 .
Fig. 9. Local detuning σ (blue), coupling strength κ (red) and the ratio σ /κ (black) as a function of void width w v for second order gratings with λ B = 1541 nm.In each case the shell width w s = Λ.In (a) the void ellipticity η v = 1 and the shell height h s = 4 µm.For (b), η v = 3 and h s = 8 µm.The dashed lines at σ /κ = 0 and −1 are a guide to the eye.