We have studied the effect of waveguide alignment on the reflection spectrum of Bragg gratings fabricated using a multiple order phase mask. We have demonstrated that splitting of certain Bragg peaks observed in earlier experiments reported in literature is caused by formation of the gratings with different periodicities in the waveguide tilted with respect to the phase mask plane due to the interference of non-symmetrical diffraction orders. Analytical expressions for spectral separation of the split peaks have been derived and verified against the experimental data recently presented in literature. The analytical predictions were also confirmed by numerical simulations of intensity distributions behind the multiple order diffraction grating and its projection on the tilted waveguide.
© 2013 Optical Society of America
First successful inscription of fiber Bragg gratings (FBGs) was reported in 1978 . Since that time several fabrication methods have been developed, including the phase mask method , interferometric method  and point by point inscription method . Although the phase mask method is most robust because of low mechanical stability requirements and has been successfully applied to fabricate gratings in silica [2, 5] and polymer fibers [6, 7], it creates certain problems related to the presence of a weak 0th diffraction order as well as higher diffraction orders of the UV inscribing beam.
In the phase masks used for gratings fabrication, typically the most efficient diffraction orders are the ± 1st. As it is shown in Fig. 1(a), the interference pattern produced by these beams has a periodicity of Λ = Λd/2, where Λd is the phase mask period. To fabricate the grating with a Bragg wavelength at λB = 1.55 μm, the phase mask period of Λd ≈1.07 μm is required according to the following relation :5, 9]. As it is shown in Fig. 1(b), this interference pattern exhibits also a periodicity in the direction perpendicular to the phase mask. One can observe interleaving planes, which replicates the periodicity of the phase mask . This effect was first described by Talbot in 1836 , while in 1881 Rayleigh calculated the period of the interference patter in z-direction, named the Talbot length. Modulation of intensity distribution in z-direction has an impact on Bragg grating reflectivity and was investigated in silica  and polymer fibers .
A presence of the Λd periodicity in the inscribed gratings was first noted by Malo et al. in 1993 . They observed additional peaks in the transmission spectrum at 1030 nm (~2λB/3) and 620 nm (~2λB/5), which cannot be explained as a reflection from the grating with Λd/2 periodicity, but could be bound to the 3rd and the 5th order reflection from the Λd grating. The phase mask they used for the grating fabrication diffracted mainly to the ± 1st orders, however, weak higher diffraction orders were also present.
In 1994 Dyer et al. suggested that the presence of the Λd/2 and Λd periodicities in the refractive index modulation combined with a complex interference pattern could be the reason for peak splitting observed when the fiber is tilted with respect to the phase mask during the inscription process . They claimed that because of a fiber tilt, a Fabry-Perot resonator is formed with maximum transmission in the center of the Bragg reflection peak, thus causing a dip in the reflected spectrum. In the next paper , Dyer et al. analyzed the intensity distribution behind the phase mask with higher diffraction orders and compared it with the structure of the grating inscribed on the surface of a polymer plate and in the optical silica fiber. They noted that due to the tilt of the phase mask, different periodicities of the grating, i.e. Λd/2 or Λd, can dominate in different segments of the fiber core and on the top surface of the polymer plate .
The effect of higher diffraction orders on the inscription process was further investigated by Smelser et al. . They studied the grating inscription with femtosecond and picosecond pulses and compared created patterns with modeled intensity distribution behind the mask. The reported results show an impact of higher diffraction orders on the grating characteristics, even if the diffraction efficiency in higher orders is below 2%.
The peaks at ~2λB/3, ~2λB/5 reported first in  were further investigated by Rollinson et al. in 2005 . The temperature and strain characteristics were measured for transmission dips at λB and ~2λB/3 to confirm their relation to the grating reflection spectra.
In the papers discussed so far [12–15,17], only two grating periodicities have been analyzed, namely Λd/2 and Λd. In 2012 Rollinson et al.  noted that the + 2nd and the −2nd diffraction orders generate the Λd periodicity due to the interference with the + 1st and the −1st diffraction orders, respectively however, they contribute also to Λd/3 periodicity (interference between the −2nd and the + 1st, and the + 2nd and the −1st) and Λd/4 periodicity (interference between the −2nd and the + 2nd diffraction orders). The complex intensity distribution behind the phase mask arising due to the interference of the 0th, ± 1st and ± 2nd diffraction orders is shown in Fig. 1(c). Rollinson et al. claimed in  that the Λd/3 and Λd/4 periodicities are unimportant because of a relatively low efficiency of higher diffraction orders and bound the observed reflection peaks at 2λB/3 and 2λB/5 with the 3rd and the 5th order reflection from the grating of period Λd. They also noted that the peaks at 2λB, 2λB/3 and 2λB/5 are sometimes split, and repeated the explanation first proposed by Yam in  that the interleaved interference planes constitute a type of π-shifted grating giving rise to double peak reflection.
The more recent work of Wade et al.  presents valuable experimental data showing the influence of possible misalignments of the inscription system on the grating characteristics. In particular, they showed that the separation of double peaks depends linearly on the tilt angle of the fiber with respect to the phase mask. They also demonstrated that in-plane rotation of the phase mask causes a shift of all the Bragg peaks. In this case, the peak splitting also changes linearly vs. a rotation angle, however, with a significantly smaller proportionality coefficient. In the discussion they noted that the explanation of the peak splitting based on π-shifted grating does not justify the observed dependence of peaks separation on the tilt angle, and more detailed analysis is needed to explain this effect.
In this work we show that splitting of certain Bragg peaks reported in several publications [5,9,19,20] is caused by formation of the gratings with different periodicities in the waveguide tilted with respect to the phase mask because of the interference of non-symmetrical diffraction orders. The proposed explanation was verified against the experimental data presented in  and showed good agreement with the results reported in this work. Without sacrificing generality, we have assumed in the presented analysis that the gratings are inscribed in the planar or strip waveguides system, which simplifies significantly the analytical and numerical calculations.
2. Influence of inscription system misalignment on Bragg peaks splitting
2.1. Waveguide tilt with respect to the phase mask
Let us first consider a situation shown in Fig. 2, in which only the 0th and the ± 1st diffraction orders are present behind the phase mask and the inscription beam is normal to the mask. This two-dimensional case is analyzed first, since tilting the waveguide in xz-plane is the only factor responsible for the Bragg peaks splitting. It will be shown in the following paragraphs that other possible misalignments, like waveguide rotation around z-axis and oblique incidence of the inscribing beam, do not cause the peaks splitting.
We select a coordinate system in such a way that the inverse vector K of the phase mask is parallel to x-axis, while the wave vector of the incidence beam k is parallel to z-axis. We also assume that the waveguide is tilted with respect to the phase mask. The directions of propagation of the diffracted waves behind the phase mask are determined by the wave vectors km, which fulfill the following relation:Fig. 2, only the 0th and the ± 1st diffraction orders are present behind the phase mask, and the waves vectors of diffracted waves can be expressed explicitly:
Orientation of the waveguide with respect to the phase mask is determined by s and t unit vectors, which are respectively tangential and normal to the waveguide surface. The vector s can be expressed as:
The s vector indicates simultaneously the direction of light propagation in the waveguide with the grating. Thus the Bragg wavelengths are determined by periodicities of the interference pattern along s direction, which can be calculated from the following relation:
In the considered case there are 3 pairs of diffracted beams, i.e.: the −1st and the + 1st orders, the −1st and the 0th orders, and the 0th and the + 1st orders. As a result, the interference pattern exhibits three different periodicities, which can be represented by the inverse vectors: k-1, + 1, k-1,0, and k0, + 1. The k-1, + 1 vector has a nonzero component only in x direction, whereas the k-1,0, and k0, + 1 vectors have also nonzero components of an opposite sign in z direction, Fig. 2. For the mode propagating in the waveguide core only the periodicities in the s-direction, determined by the scalar products of the k-1, + 1, k-1,0, and k0, + 1 vectors with s, cause the Bragg reflections.
If the waveguide is parallel to the phase mask, the following relation holds:Eq. (8) is no longer fulfilled. In this case, each pair of interfering beams produces different periodicity along the direction of propagation of the core mode represented by the vector s:
Applying Eq. (1), one can calculate the dependence of the Bragg wavelength upon a tilt angle. For the peak at λB, originating from the interference of the −1st and the + 1st diffraction orders, one obtains:
The interference between the −1st and the 0th diffraction orders results in Λ-1,0 periodicity, which is slightly different than Λ0, + 1 corresponding to the interference of the + 1st and the 0th orders. As a result, two reflection peaks located close to 2λB will be observed, namely:
The separation between the two peaks Δλ(φ) = λ0, + 1(φ) – λ-1,0(φ) is given by the formula:
Disregarding the dispersion of the mode effective index leads to the following expression:
2.2. General case
Let us consider now the most general three dimensional case, in which the inscribing beam is not normal to the phase mask, presented in Fig. 3. Assuming that the incidence plane makes an angle β with xz-plane and the incidence angle is α, one can express the wave vector k of the inscribing beam as:
In the most general case, the waveguide misalignment can be described with two angles. As in the previous case, φ represents the tilt angle of the waveguide with respect to the phase mask (angle between s vector and xy-plane), while θ is the waveguide rotation angle with respect to z-axis. In such a case, the s unit vector indicating the direction of mode propagation can be expressed as follows:
The periodicity along s direction, which originates from the interference of mth and qth diffraction orders, can be determined using Eq. (7):
Equation (24) shows that in the waveguide parallel to the phase mask (φ = 0) the following periodicities are present in the grating:
If the waveguide is not perfectly parallel to the phase mask (φ ≠ 0), then the peaks’ splitting arises since Λm,q and Λq,m are different according to Eq. (24). Moreover, multiple splitting of certain peaks can be observed because Λm,q corresponding to the interference of mth and qth diffraction orders having the same difference (m −□q) are not equal. For example, if the 0th, the ± 1st and the ± 2nd diffraction orders are present behind the phase mask, then not only two gratings with periodicity close to Λd will be inscribed as a result of the interference of the −1st and the 0th orders, and the 0th and the + 1st orders but also additional two gratings due to the interference of the −2nd and the −1st orders, and the + 2nd and the + 1st orders. As a result, the Bragg peak at 2λB is split into four peaks corresponding to the grating periodicities Λ+2, + 1, Λ+1,0, Λ0,-1, Λ-1,-2. In the experiments reported so far, only the peaks corresponding to Λ+1,0 and Λ0,-1 have been observed because the Talbot length for the interference pattern created by the −2nd and the −1st orders, and the + 2nd and the + 1st orders is usually much smaller than the core diameter. As a result, the refractive index modulations with periodicities Λ+2, + 1 and Λ-1,-2 are averaged over the fiber core cross section. Moreover, our analysis points to the existence of the side-peaks corresponding to Λ+2,0 and Λ-2,0, which move away from the peak at λB (created by periodicity Λ-1, + 1) with an increasing tilt. Similarly as in the previous case, the existence of the tilt-sensitive side-peaks at λB has not been yet confirmed experimentally because the Talbot length of the pattern created by the interference of the ± 2nd and the 0th orders is much smaller than the fiber core cross section.
3. Numerical simulations
The results presented so far allow to predict the refractive index periodicities inscribed in the waveguide core, however, do not tell anything about the corresponding modulation depth of the effective index of the guided mode. In order to estimate the relative amplitudes of effective index modulation, the numerical simulations were conducted. After  we assumed the following parameters in the simulations: Λd = 536 nm, the wavelength of the inscribing beam λUV = 244 nm, and the material refractive index n = 1.511 at λUV. Moreover, the diffraction efficiencies were as follows: 3.0%, 31.2% and 17.3% respectively for the 0th, ± 1st and ± 2nd diffraction orders. Subsequently, the interference pattern in the core of the planar waveguide was calculated taking into account the Fresnel reflection from its top surface.
Three cases were numerically analyzed assuming normal incidence of the inscribing beam: the phase mask parallel to the waveguide with the mode field diameter equal to 4 μm, Fig. 4(a); the same waveguide tilted with respect to the phase mask by the angle φ = 0.1°, Fig. 4(b); and the waveguide with the mode field diameter of 1 μm tilted by the angle φ = 0.1°, Fig. 4(c). The results of numerical simulations presented in Fig. 4 are in agreement with analytical formulas. Tilting of the waveguide leads to splitting of the selected Bragg peaks because additional periodicities occur in longitudinal modulation of the mode effective index. For a typical value of the mode filed diameter (>3 μm) one can observe splitting of the 1/Λd, and 3/Λd spatial frequency into two peaks. For very thin waveguides (mode field diameter 1 μm) our simulation results predict the appearance of four split peaks. For the phase mask with greater number of diffraction orders even more periodicities may be present in the waveguide core. However, because of averaging over the mode diameter greater than Talbot length, those periodicities do not produce peaks in the reflection spectrum. Moreover, the numerical results presented in Fig. 4 confirm the existence of the weak side peaks near the strong peaks located at 1/Λd and 2/Λd. The separation of the side-peaks is proportional to the tilt angle and they overlap with the central peaks when the waveguide is parallel to the phase mask.
To independently confirm the results based on the Fourier analysis, we performed the simulations of reflection spectrum for the mode propagating in the planar waveguide with an inscribed grating. Using a finite element method solver (COMSOL Multiphysics – Wave Optics Module), we modeled the fundamental mode propagation through the grating inscribed in 1 μm thick silica planar waveguide surrounded by air. The simulations were carried out at around of 2λB ≈1.453 μm. The refractive index of silica for this wavelength is 1.445, which results in an effective mode index equal to 1.357. The calculated mode field diameter is 1.15 μm at 1.453 μm and the grating length assumed in the simulations is Λd × 212 ≈2.20 mm. The mesh of the finite element method consists of 570 000 elements. The parameters describing the phase mask are the same as for the Fourier analysis. The interference pattern behind the phase mask was converted into refractive index modulation in the waveguide core with an amplitude of 10−4. The calculated spectral dependence of the reflection coefficient is presented in Fig. 5. There are clearly visible four Bragg peaks corresponding to four different pairs of diffraction orders in the reflection spectrum. The peaks’ positions and relative heights are in good agreement with the results of the Fourier analysis presented in Fig. 4(c).
4. Comparison with experimental data
Finally, we verified the predictions of our model against the experimental results reported by Wade et al. in . They investigated splitting of the peak at λ(2) = 1555 nm (close to 2λB) obtained in the optical fiber using the phase mask of period Λd = 536 nm and the inscribing beam λUV = 244 nm. The coefficient κ calculated according to Eq. (10) takes the value of 0.2408, which according to Eq. (19) gives the proportionality coefficient between the peaks separation and the tilt angle equal to 2λ(2)κ = 13.1 nm/deg, whereas the coefficient determined in the experiment is 15.8 nm/deg. Similarly, we estimated after  the separation of Bragg peaks at λ(2/3) = 1036 nm (close to 2λB/3) obtained for the phase mask period Λd = 1.0703 μm. Assuming that this peak originates from the 3rd order reflection from the grating of periodicity Λ+1,0, one gets the proportionality coefficient between the tilt angle and the peaks separation equal to 4.18 nm/deg. If the peaks result from the 1st order reflection from the grating with periodicities Λ+1,-2 and Λ-1, + 2, the proportionality coefficient will be equal to 4.42 nm/deg. The two values are close to the coefficient obtained in the experiment equal to 5.4 nm/deg.
The discrepancies between the calculated and measured coefficients for 2λB and 2λB/3 are of the same order and are most probably caused by the same physical reasons. In particular, the derived analytical formulas for peaks splitting and the numerical simulations are strictly valid for planar and strip waveguides, while the experiments were conducted for cylindrical waveguides. This may be the reason for observed discrepancies because the fiber curvature was not accounted for in the presented analysis. The second possible reason could be focusing of the UV beam on the fiber during the grating inscription, while it is assumed in the model that the inscribing beam is collimated and can be represented by one wave vector k. It should be stressed, however, that although the geometry of the inscribing system may influence slightly the proportionality coefficient between the tilt angle and the peaks separation, it does not change the physical reasons behind the splitting effect. Therefore, the qualitative conclusions of the analysis presented in this work are valid for any geometry of the waveguide and the inscribing system.
We also tried to explain the results reported in  for the phase mask rotation around z-axis. For normal incidence of the inscribing beam, the peak splitting by 0.94 nm observed in the experiment is equivalent to the tilt of the phase mask by the angle of ~0.072°. According to Eq. (24), rotation of the phase mask by 4° changes the peak separation from 0.9411 nm to 0.9458 nm. This value is much smaller than 1.16 nm observed in the experiment. It is probable that in the reported experiment, the rotation axis was not perpendicular to the phase mask plane. In such a case, rotation of the phase mask changes the tilt angle between the mask and the waveguide. To increase the peaks separation up to ~1.16 nm, the required tilt angle change is only 0.016°. It is possible to induce such a tilt angle change with 4° phase mask rotation if the rotation axis is deviated by 0.24° from normal direction.
We have proposed a new consistent model explaining Bragg peaks splitting in the gratings fabricated using a multiple order phase mask. Analytical and numerical studies show that the effect is caused by the tilt of the waveguide with respect to the phase mask, which gives rise to different periodicities of refractive index modulation associated with the interference of non-symmetrical diffraction orders. The proposed model predicts the linear dependence of the peaks splitting upon the tilt angle, which is in agreement with recent experimental observations . The proportionality coefficients between the calculated peaks separation and the tilt angle are by about 20% lower than the experimental values. The observed discrepancy can be caused by different symmetry of the waveguides used in the experiment and/or by different alignment of the inscribing system than assumed in our model. We also showed that other possible misalignments of the inscription system like the oblique incidence of the inscribing beam and the rotation of the waveguide in the mask plane do not produce any Bragg peaks splitting, however, may change the proportionality coefficient between the peaks separation and the tilt angle.
It is worth to underline that for the phase mask with multiple diffraction orders, our model predicts tilt-induced splitting into a greater number of peaks located symmetrically with respect to 2λB, 2λB/3 and 2λB/5, etc., arising due to the interference of higher order non-symmetrical pairs of diffracted beams. So far only the peaks produced by the interference of non-symmetrical low order beams were observed experimentally. For higher order beams the Talbot length is so short that the corresponding index modulation is averaged over the waveguide core. Moreover, our analysis points to the presence of the weak side-peaks near the strong peaks located at 2λB and λB, etc., with the separation of the side-peaks being proportional to the tilt angle. Again, the existence of the side-peaks has not been yet confirmed in the experiment due to short Talbot length of the corresponding interference patterns.
The work was supported by Wroclaw Research Center EIT + within the project “The Application of Nanotechnology in Advanced Materials” – NanoMat (POIG.01.01.02-02-002/08) co-financed by the European Regional Development Fund (Operational Programme Innovative Economy, 1.1.2). K. Tarnowski acknowledges support of the Foundation for Polish Science START Program.
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