Cladding mode coupling in highly localized fiber Bragg gratings II: complete vectorial analysis

doi:10.1364/OE.20.021434

Cladding mode coupling in highly localized fiber Bragg gratings II: complete vectorial analysis

Jens U. Thomas, Nemanja Jovanovic, Ria G. Krämer, Graham D. Marshall, Michael J. Withford, Andreas Tünnermann, Stefan Nolte, and Michael J. Steel »View Author Affiliations

Optics Express, Vol. 20, Issue 19, pp. 21434-21449 (2012)
http://dx.doi.org/10.1364/OE.20.021434

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▸ Article Outline
– Abstract
1. Introduction
2. Modeling spectral properties of localized FBG
3. Imaging of the modes
4. Classification of the cladding mode reflections
5. Conclusion and outlook
– References and links

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Abstract

Highly localized fiber Bragg gratings can be inscribed point-by-point with focused ultrashort pulses. The transverse localization of the resonant grating causes strong coupling to cladding modes of high azimuthal and radial order. In this paper, we show how the reflected cladding modes can be fully analyzed, taking their vectorial nature, orientation and degeneracies into account. The observed modes’ polarization and intensity distributions are directly tied to the dispersive properties and show abrupt transitions in nature, strongly correlated with changes in the coupling strengths.

1. Introduction

A fiber Bragg grating (FBG) is a periodic modification of the fiber core that converts co-propagating modes into counter-propagating ones. The modes do not have to be similar, e.g. an FBG can be used to convert the co-propagating fundamental mode into a counter-propagating higher order mode [1

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997). [CrossRef]

–3

C. Lu and Y. Cui, “Fiber bragg grating spectra in multimode optical fibers,” J. Lightwave Technol. 24, 598–604 (2006). [CrossRef]

]. The FBG can also convert different higher order modes [4

D. Sáez-Rodriguez, J. L. Cruz, A. Díez, and M. V. Andrés, “Coupling between counterpropagating cladding modes in fiber Bragg gratings,” Opt. Lett. 36, 1518–1520 (2011). [CrossRef] [PubMed]

]. In a single mode fiber, only the fundamental mode is guided within the core and all higher order modes are supported by the cladding. An FBG inscribed in such a fiber thus causes multi-resonant coupling of the core mode into counter-propagating cladding modes [5

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997). [CrossRef]

] as well as inter modal coupling of cladding modes [4

D. Sáez-Rodriguez, J. L. Cruz, A. Díez, and M. V. Andrés, “Coupling between counterpropagating cladding modes in fiber Bragg gratings,” Opt. Lett. 36, 1518–1520 (2011). [CrossRef] [PubMed]

]. The transmission spectrum of the core mode hence exhibits a series of dips on the short wavelength side of the Bragg resonance. FBGs with strong cladding mode resonances have found widespread applications in sensing [6

T. Guo, C. Chen, and J. Albert, “Non-uniform-tilt-modulated fiber bragg grating for temperature-immune micro-displacement measurement,” Meas. Sci. Technol. 20, 034007 (2009). [CrossRef]

–8

A. Martinez, Y. Lai, M. Dubov, and I. Khrushchev, “Vector bending sensors based on fibre bragg gratings inscribed by infrared femtosecond laser,” Electron. Lett. 41, 472–474 (2005). [CrossRef]

]. An important example is refractive index sensors, which exploit the fact that cladding modes of high radial order are sensitive to the exterior of the fiber. On the other hand, cladding modes of higher azimuthal order are needed for directional sensing of fiber bends and twists. The latter can only be achieved with FBGs that have an asymmetric cross-section. Until now, this was commonly achieved by tilting the FBG in the fiber core [2

K. Lee and T. Erdogan, “Fiber mode conversion with tilted gratings in an optical fiber,” J. Opt. Soc. Am. A 18, 1176–1185 (2001). [CrossRef]

In a recent publication we demonstrated that exceptionally strong coupling to cladding modes can also be realized using highly localized FBGs [9

J. U. 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). [CrossRef] [PubMed]

]. In this context, highly localized means that the index perturbation covers only a fraction of the core and the index contrast is high (of the order of several tens of percent). FBGs with such properties can be written using the point-by-point (PbP) process with tightly focused ultrashort laser pulses: state-of-the-art setups allow for inscription of periodic chains of micrometer sized voids within the core of the fiber [10

G. Marshall, R. 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). [CrossRef] [PubMed]

,11

R. J. Williams, C. Voigtländer, G. D. Marshall, A. Tünnermann, S. Nolte, M. J. Steel, and M. J. Withford, “Point-by-point inscription of apodized fiber bragg gratings,” Opt. Lett. 36, 2988–2990 (2011). [CrossRef] [PubMed]

]. In this previous work [9

], we exploited the fact that highly localized FBGs exhibit a fundamentally different coupling behavior as compared to common non-tilted, homogeneous FBGs: while the latter allow only coupling to and from azimuthally independent fundamental fiber modes [5

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997). [CrossRef]

], highly localized FBGs grant efficient access to fiber modes of higher azimuthal order. We investigated the possibilities of exciting different classes of modes, and showed that the spectral response could be reproduced numerically using the analytic expressions for a circular fiber, which incorporate both the core-cladding as well as the cladding-exterior boundary [5

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997). [CrossRef]

, 12

C. Tsao, D. Payne, and W. Gambling, “Modal characteristics of three-layered optical fiber waveguides: a modified approach,” J. Opt. Soc. Am. A 6, 555–563 (1989). [CrossRef]

However, there are two major features hidden within the resonances of the spectrum. Firstly, modes of higher azimuthal order are degenerate to lower ones—at a single resonance, several cladding modes can be excited. Secondly, higher order modes are no longer linearly polarized and their reflected field profiles can be rotated with respect to the incident core mode polarization. These features can be revealed by imaging the reflected cladding modes. Thus, the goals of this paper are as follows: to determine the degeneracies of the cladding modes within the three-layer model, to compute the strength of the multi-mode reflection and to eventually predict the resulting fields at each resonance. This also serves as a starting point towards the ultimate goal of precise excitation of any desired cladding mode. The paper is structured as follows: In Sec. 2, the vector modes of the cladding and their fundamental coupling behavior are reviewed. As in [9

], our emphasis is on distinguishing between features that result from the modal structure of the fiber and those which are set by the position of the highly localized FBG. The imaging of the cladding modes is described in Sec. 3. Section 4 provides a detailed discussion of the reflected patterns and the comparison with the semi-analytical model. We show that the reflected modes can be classified in distinct families and transitions from one family to the next are intimately connected to changes in the dispersion properties we term “virtual cutoffs” [9

2. Modeling spectral properties of localized FBG

2.1. Vector modes of the three-layer fiber

To understand the rather complex structure of the reflection spectra for an FBG with an arbitarily shaped cross-section, we need to briefly recall some properties and notations for the vectorial modes of the three-layer circular step-index fiber [12

C. Tsao, D. Payne, and W. Gambling, “Modal characteristics of three-layered optical fiber waveguides: a modified approach,” J. Opt. Soc. Am. A 6, 555–563 (1989). [CrossRef]

]. The modes are labeled with azimuthal (l) and radial (m) indices, such that all field components are proportional to exp(iϕl)exp(iβz) in cylindrical coordinates (r, ϕ, z), with l = 0, 1, 2.... The radial index m numbers all solutions for a given l, starting with m = 1 for the solution with the highest effective refractive index n̄ = βλ/(2π), which depends on the wavelength λ of the mode and its propagation constant β(λ). The fields of modes with l = 0 are azimuthally invariant and either purely azimuthally or radially polarized. The electric field of an azimuthally polarized mode is always parallel to a cylindrical surface. Thus the electric field has no z-component and such modes are transverse electric (TE). The same holds for the magnetic field of fully radially polarized modes, which are transverse magnetic (TM). In contrast, modes with l > 0 always have longitudinal electric and magnetic field components. They are designated hybrid modes and are classified EH or HE, according to whether the electric or magnetic components make the dominant contribution to the longitudinal field [13

E. Snitzer, “Cylindrical dielectric waveguide modes,” J. Opt. Soc. Am. 51, 491–498 (1961). [CrossRef]

]. In the standard notation, the EH and HE radial modes for given l are numbered separately. In the present case where we need to refer to sequences of resonant lines in transmission spectra, it is more convenient to use a single index m to index all modes of fixed l, and simply note which are of EH or HE character when needed.

Just as in the two-layer model, the fundamental HE₁₁ mode is linearly polarized and has the highest n̄. As in [9

], we have used Tsao’s expressions [12

C. Tsao, D. Payne, and W. Gambling, “Modal characteristics of three-layered optical fiber waveguides: a modified approach,” J. Opt. Soc. Am. A 6, 555–563 (1989). [CrossRef]

] to compute the effective refractive indices n̄_lm(λ), and their corresponding electric and magnetic fields E and H. All expressions for the hybrid mode fields and dispersion relation are given in the appendix of [9

], while TE and TM mode expressions are provided in appendix A of this paper. For given l the EH and HE modes appear in near-degenerate pairs. For l ≥ 1, there is a further exact degeneracy. Because of the rotational symmetry of the fiber, each hybrid mode l, m has a degenerate orthogonal counterpart whose fields are rotated by π/2l. These are designated as “even” or “odd” modes as in Ref. [14

A. W. Snyder and W. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978). [CrossRef]

]. For the fundamental HE₁₁ mode these terms correspond to the axis of polarization being along the x- or y-axis, respectively.

2.2. Cladding mode resonances and degeneracies

An FBG with period Λ in a single mode fiber couples the fundamental HE₁₁ mode into various cladding modes at different wavelengths. Each of these resonances can be computed by self-consistently solving the generalized Bragg equation [9

]

λ = ({\bar{n}}_{11} (λ) + {\bar{n}}_{l m} (λ)) Λ / ν

(1)

for the relevant Fourier order ν (see section 2.3). Figure 1 displays a typical spectral distribution of such cladding mode resonances. They have been computed for a single mode step-index fiber in the telecommunication regime (core radius a₁ = 4.15 μm, cladding radius a₂ = 62.5 μm, refractive indices n₁ = 1.4670, n₂ = 1.4618 and n₃ = 1.0 (air) respectively). The grating period is Λ = 1.062 μm and the resonances have been computed for ν = 2.

Fig. 1 Spectrally sorted resonances of the investigated FBG. Coupling constants have been evaluated for the transverse geometry shown in the inset. The polarization direction of the launched fundamental mode is indicated by the double arrow. In the graph, the height of the red lines represents the coupling strength of the fundamental mode to TE or HE modes, the blue lines for TM or EH modes. For HE and EH modes, the line represents the sum of the coupling constants for even and odd modes. The horizontal dash separates their individual contributions, with the lower part representing coupling to the odd mode. The virtual cutoff wavelengths separating the different coupling regimes (see [9

]) are shown as vertical dashed lines. The ℓ = 1 and ℓ = 2 degeneracies are highlighted at 1540 nm by two vertical dotted lines.

Figure 1 is separated into individual plots for the TM/TE modes and the EH/HE mode families for l = 1...4. The radial index m increases from right to left. Starting with l = 0 at the bottom of the graph, the resonances are plotted up to l = 4. The wavelength of each resonance is represented by the colored lines, whose length corresponds to the coupling strength (which will be discussed below). Note that the l ≥ 1 resonances always appear in near-degenerate doublets, with the shorter wavelength mode being of EH character and the longer wavelength being HE. In many cases, the degeneracy is too close for the two modes to be resolved on this scale. The l = 0 resonances also occur in doublets, with the TM mode on the shorter wavelength side and TE on the longer.

Figure 1 reveals a further near degeneracy: doublets with l = 1, 3, 5,.. are found at virtually the same wavelength (the difference can not be resolved on this scale, since Δn̄ < 1×10⁻⁶) and in between these combs, the TM/TE pairs are degenerate with the l = 2, 4, 6,... hybrid modes. (This fact is immediately apparent if Fig. 1 is viewed by the reader at a oblique angle from below but we also provide a visual aid with two vertical dotted lines at 1540 nm.) The existence of these degeneracies is important, because it implies that the spectrum alone does not yield enough information to identify which azimuthal modes have been excited at a given resonance and that the specific field overlaps with the index modification become critical. In the following we label these degenerate families of modes with a calligraphic symbol ℓ = 1 and ℓ = 2, where this index stands for all modes with l being odd or even, respectively.

The high degree of degeneracy does not necessarily mean that all degenerate ℓ modes are excited at a given resonance. Efficient coupling only occurs to a limited number of cladding modes. In general, the coupling efficiency

κ_{l m} = \frac{2 π c}{4 λ} \cdot \int_{0}^{2 π} d ϕ \int_{0}^{a_{1}} d r r Δ ε (x, y, z) E_{11}^{T} \cdot E_{l m}^{T *},

(2)

depends both on the perturbation of the dielectric constant Δε (r,ϕ, z) (provided by the FBG) and the transverse electric fields of the incident fundamental mode

E_{11}^{T}

and reflected mode

E_{l m}^{T *}

of the fiber. The radial integral is restricted to the core since our grating modifications are always localized in the core. Note that the contribution of the longitudinal field components to the coupling constants are negligible in this work and have been dropped. Also note that the polarization orientation of the incident fundamental mode is included in

E_{11}^{T}

, which is the superposition of even and odd HE₁₁ modes. In the following the polarization is always oriented along the y-axis, thus only the odd HE₁₁ mode is incident.

2.3. Index modification geometry

In the remainder of the paper, we investigate the coupling due to micro-voids written PbP using an ultrafast laser source. From microscope images, the shape of the micro-voids in our FBGs were determined to be approximately ellipsoidal, with a width of w = 0.4 μm and a height of h = 1.9 μm. The refractive index of the voids is assumed to be 1 [15

N. Jovanovic, J. U. 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). [CrossRef] [PubMed]

]. They are surrounded by an elliptical region of densified material. We neglect any asymmetries of the void and its densified shell [15

], accepting that the resulting coupling constants may be systematically too small. Thus, the transverse refractive index profile of the micro-void Δn_max(x, y) = (n₁ − 1)θ(x, y) is approximated by a homogeneous ellipse with the aforementioned dimensions, where θ(x, y) = 1 inside the ellipse and 0 elsewhere. A sketch of the core cross-section is depicted in the inset of Fig. 1. The grating itself is treated as a periodic square modulation along the fiber axis, which allows the approximation

Δ ε (x, y, z) \approx 2 ε_{0} n (x, y) Δ n (x, y, z) = ε_{0} n (x, y) (Δ n_{0} (x, y) + \sum_{ν = 1}^{\infty} Δ n_{ν} (x, y) 2 cos (\frac{2 π ν}{Λ} z)),

(3)

with the Fourier expansion coefficients evaluated to be

Δ n_{0} (x, y) = \frac{2 w}{Λ} Δ n_{max} (x, y) for the “ DC ” portion and

(4a)

Δ n_{ν} (x, y) = \frac{2}{π ν} sin (π ν \frac{w}{Λ}) Δ n_{max} (x, y)

(4b)

for the “AC” portions. The different Fourier orders ν = 1, 2, 3... usually do not occur within the same spectral region. (Though in some cases we have observed 1^st order coupling to very high (m ≈ 1000) cladding modes right up to the 2^nd order fundamental Bragg resonance [9

]). In the following we concentrate on second order (ν = 2) resonances, because the micro voids would overlap for the 0.5 μm period necessary for a 1^st order Bragg wavelength within the near infrared.

2.4. Virtual cutoffs and changes in coupling strength

We now examine the dependence of the coupling strength (Eq. (2)) on the mode order. In Fig. 1, the amplitude of the computed coupling constants κ_lm is represented by the height of the blue (EH and TM) and red (HE and TE) lines. For l > 0, the contribution from the even mode coupling coefficient is plotted on top of the odd mode coefficient, separated by a short horizontal dash. The ratio of even and odd modes determines the field orientation as we discuss in Sec. 2.5.

We observe a strong dependence on l, m, the EH/HE character and the even/odd character of the mode. Note for example, that significant coupling to the HE modes occurs for lower values of m than for the EH modes. Here we summarize the influences that give rise to these dependencies. Full details are provided in Ref. [9

One prerequisite for efficient coupling with intra-core fiber gratings is that

E_{l m}^{T *}

is non-zero within the core region. While this requirement is met by all fundamental HE_1m modes, EH_1m modes only have a field contribution at the core when their radial index m is sufficiently high. In our previous study, we introduced the concept of a “virtual cut-off” to help account for the changes in coupling strength with l and m and the EH/HE nature of the mode. At each virtual cutoff U(l, m′), the number of intensity rings within the core increases by one [9

]. Specifically, U(l, m′) is the lowest m value for which HE_lm has m′ rings within the core. (This definition is equivalent to a slightly different form used in Ref. [9

]).

The character of the modes changes qualitatively at the first few virtual cutoffs. For modes with m < U(l, 1′) there is essentially no intensity within the core, and the polarization is uniform. Modes falling in the range U(l, 1′) < m < U(l, 2′) have one maximum within the core if they are of HE character, whereas EH modes in this range have a minimum at the center and significant field contributions only near the core-cladding boundary. The field orientation varies in a complex fashion across the mode profile, and both EH and HE modes have comparable fractions of energy in the H_z and E_z components. In contrast, for m > U(l, 2′), the energy in the longitudinal components resides almost purely in H_z (HE) or purely in E_z (EH). Therefore, the transverse field of the HE modes eventually becomes close to purely azimuthally polarized (quasi-TE) and that of the EH modes becomes strongly radially polarized (quasi-TM).

In Fig. 1, virtual cut-off wavelengths (computed from the corresponding n̄ and the period Λ) are plotted as vertical black dashed lines, labeled with U(l, m′). For a given azimuthal index l, the first “virtual cut-off” U(l, 1′) marks the onset of coupling to the HE modes, while for m > U(l, 2′), EH modes couple as well.

The most distinct feature of the highly localized FBGs is that the coupling properties are more determined by the transverse position of the index modifications within the core than by their shape. For certain positions of the FBG within the core, positive and negative contributions of the coupling integral (Eq. (2)) cancel out. For example, a well-centered FBG only supports coupling to l = 1 modes, as does a conventional transversely homogeneous FBG. In fact, coupling to l > 1 modes is only possible if Δε is inhomogeneous. Thus, excitation of ℓ = 2 resonances is clear evidence for higher order mode coupling. The farther from the center the modification is placed, the stronger the coupling to higher l modes becomes [9

]. However, an upper limit for coupling to higher l modes is set by the virtual cut-offs—there must be significant mode energy within the core.

2.5. Role of polarization

The evolution in mode polarization with m also affects the structure of the coupling coefficients in Fig. 1. For m < U(1, 2′), the HE_1m modes are purely linearly polarized. This means that with the choice of an odd HE₁₁ launched core mode (see inset of Fig. 1), the coupling coefficient is always zero for even HE_1m modes.

In general, there can be coupling to both the even and odd HE_lm modes with the net effect that the reflected mode pattern is rotated with respect to the input mode orientation. The azimuthal position of the FBG with respect to the incident polarization determines the degree of rotation of the reflected mode. In Ref. [9

], we evaluated the case that the micro-void is only displaced perpendicular to the axis of polarization. Consequently, positive and negative contributions of the even coupling integral cancel out, and the FBG couples only to odd cladding modes. However, for the FBG investigated in the following, we determined a displacement of d_x = 0.8 μm and d_y = 0.7 μm from microscope images. Therefore, both even and odd orientation modes are excited and we expect to observe rotational effects in the images of the reflected fields.

In conclusion, by knowing the fiber geometry and the resulting propagation constants of the cladding modes, each peak in the spectrum can be attributed to one or more azimuthal indices l, which are either all odd or even integers. Since these sets of higher order modes are degenerate, more specific conclusions cannot be drawn from the spectrum alone. In this context, the virtual cut-offs are useful because they set an upper limit to l. For example, for the wavelength range of our recorded spectra, cladding modes with l > 6 carry no power within the core and resonant coupling to them can be neglected. Thus, a peak can either be a superposition of modes with l = 1, 3, 5 or l = 0, 2, 4, 6. The selection can be further narrowed down if the cross-section of the index modifications is known reasonably well. From the coupling integrals of the investigated FBG, efficient coupling is only expected for azimuthal orders l ≤ 4. Another quantity not accessible through the spectrum alone is the orientation of the reflected mode with respect to the plane of polarization of the incident core mode. In the next sections, we show how these remaining ambiguities can be resolved by imaging the reflected modes and characterizing their polarization states.

3. Imaging of the modes

The investigated FBG was inscribed in standard single mode fiber (Corning SMF-28e) by ultra-short pulses. It is composed of a chain of micro-voids within the core, separated by a longitudinal period of Λ = 1.062 μm, corresponding to a 2^nd order fundamental-mode Bragg resonance at λ = 1554.5 nm. More details of the fabrication process are given in [9

,10

,15

]. For measuring and imaging the cladding modes, a setup similar to that reported by Eggleton et al. [16

B. Eggleton, P. Westbrook, C. White, C. Kerbage, R. Windeler, and G. Burdge, “Cladding-mode-resonances in air-silica microstructure optical fibers,” J. Lightwave Technol. 18, 1084–1100 (2000). [CrossRef]

] was used (see Fig. 2). The source was a swept wavelength system (SWS), that could be tuned from 1520 nm to 1570 nm with a line-width of 2 pm (JDS Uniphase). The light was launched with a polarization maintaining (PM) fiber, providing a linearly polarized beam. The imaging setup is shown in Fig. 2. The light was collimated and passed through a beam splitter before being coupled into the core of the fiber containing the FBG by a 40× objective lens. The fiber had been cleaved such that the FBG was located adjacent to the end facet of the fiber. The light transmitted through the fiber core was recorded with a photodiode. Sweeping the laser wavelength thus provided a full transmission spectrum.

Fig. 2 Setup for imaging the cladding modes for PbP FBGs. Transmission spectra were also obtained by recording the signal of the photodiode while sweeping the wavelength of the SWS.

The light reflected from the FBG, in either core or cladding modes, was collimated by the same objective lens used to couple the light into the fiber, and then partially reflected by the beam splitter before being imaged onto an IR sensitive camera (Vidicon). The wavelength of the SWS was tuned to each resonant wavelength and its corresponding reflected mode pattern was recorded. For each observed mode, the polarization was investigated by inserting a polarizer before the IR camera, and studying the change in the mode pattern as the polarizer was rotated. The polarization was accordingly classified as linear, azimuthal, radial or spatially non-uniform.

The measured reflection spectrum is shown in Fig. 3. All the resonances could be reproduced numerically by solving the hybrid mode dispersion relation for l = 1 and l = 2 [9

]. The resonances are labeled in Fig. 3 by their l and m labels (similar to Fig. 1). For each resonance doublet, the longer wavelength resonance corresponds to coupling to the hybrid HE mode, while the shorter wavelength is the hybrid EH mode. Both HE and EH modes are consecutively numbered with the single index m. For clarity, only the HE modes are labeled with (ℓ, m) in Fig. 3. In each doublet, the peak on the shorter wavelength side is the EH mode (ℓ, m + 1).

Fig. 3 Typical transmission spectrum with lowest azimuthal order vectorial labels (l, m). All numbers refer to the HE resonances, which are always at the longer wavelength of the EH/HE doublets. Red labels denote ℓ = 1 modes, blue labels denote ℓ = 2 modes, the vertical dotted lines correspond to the computed l = 1 and l = 2 resonances of the hybrid modes (see also Fig. 1). Horizontal lines below the spectrum indicate the range over which mode patterns of various forms were observed. The virtual cutoffs are also labeled. The lower plots are simply magnifications of the main plot.

A selection of typically observed intensity patterns for this grating is depicted in Fig. 4. In the following we qualitatively refer to the mode patterns as “rings” (Fig. 4(a)), “bow ties” (Fig. 4(b)) and “quad ties” (Fig. 4(c) and 4(d)). These terms are only descriptive and there are many underlying mode compositions yielding the same number of lobes. This will be explained in more detail in Sec. 4.

Fig. 4 Typical mode patterns observed of each class: (a) ring, (b) bow tie, (c) and (d) quad tie. Labels indicate the mode indices and the wavelength at which they were observed.

The spectral position of the resonance determines the type of mode pattern observed. These observations are summarized with the horizontal lines in Fig. 3. In the following, we discuss several representative modes, since the mode patterns of a given resonance group are similar (only the number of rings differs).

We can summarize our observations of the mode patterns and polarizations as follows. A full explanation of these observations is provided in the next section. Close to the Bragg peak, the modes are all fully linearly polarized in the same direction as the incident laser along the horizontal y-axis (which is parallel to the long axis of the micro void). While the ℓ = 1 patterns have a ring-like intensity pattern (Fig. 4(a)), the ℓ = 2 have more of a bow-tie distribution. Rotating the launch state of polarization by 90 degrees transforms a vertically polarized mode into a horizontal one with the spatial distribution conserved.

The ring distribution of the ℓ = 1 becomes bow-tie shaped below 1550 nm, coinciding with the predicted location of the U(1, 2′) virtual cutoff. Approximately 12–15 nm from the Bragg peak, double peaks begin to be visible for the ℓ = 1 states. At each double peak, the mode pattern at the shorter wavelength is observed to be predominantly radially polarized, while the longer wavelength mode pattern is predominantly azimuthally polarized. In that regime, the ℓ = 1 and ℓ = 2 mode patterns both have a bow-tie intensity pattern. The HE and EH bow-tie are oriented 90° to each other. For the bow-tie modes the spatial orientations of the bow-ties in the doublet swaps if the polarization of the incident light is rotated by 90 degrees. In that case the longer wavelength peak becomes the horizontal bow-tie and the shorter wavelength peak becomes the vertical bow tie. Further from the Bragg peak and below the U(1, 4′) cutoff at 1532 nm, the ℓ = 1 mode pattern develops a more complex fourfold quad-tie structure. In this wavelength range, a distinct axis of polarization cannot be assigned anymore.

4. Classification of the cladding mode reflections

In the preceding section we observed that major changes of the intensity distribution and the polarization of the reflected cladding modes coincide with the virtual cut-offs. Now we explain this relation in more detail. In the following we divide the problem according to the observed polarization states of the reflected mode patterns: a linear polarized (LP) regime, a (quasi-) TE/TM regime, and a regime where the polarization is more complex due to contributions of cladding modes of higher azimuthal order.

All representative mode patterns were reproduced with the following procedure. As discussed above, the cladding modes often have degeneracies, so that multiple cladding modes are usually excited at a single resonant wavelength. We therefore rely on the coupling constants κ_lm from Sec. 2.2, in order to predict their relative amplitudes within the reflected patterns. Unfortunately, analytic expressions do not exist for multimode reflections and it is necessary to solve the differential equations of the coupled mode system numerically [5

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997). [CrossRef]

]. However, only the peak reflectivity of the individual modes is needed to predict the reflected pattern, so these equations can be greatly simplified. Given a resonance with N degenerate modes, the detuning (the phase mismatch) can be assumed to be zero for each mode, which reduces the problem of a set of N + 1 coupled mode equations for the complex amplitudes A₁₁ of the co-propagating core mode and the counterpropagating cladding modes B_lm to the system

\frac{d A_{11} (z)}{d z} = \sum_{l, m} - i κ_{l m} B_{l m} (z)

(5)

\frac{d B_{l m} (z)}{d z} = i κ_{l m} A_{11} (z) .

(6)

This set of ordinary differential equations. then has to be solved with the boundary conditions A₁₁(z = −L/2) = 1 and B_lm(z = +L/2) = 0. The fraction of light reflected in the cladding mode (l, m) is R_lm = |B_lm(z = −L/2)|². Intermodal coupling through fiber bends or propagation-related phase changes can be neglected because the FBG sits at the end of the fiber where the light is coupled in. The amplitudes B_lm allow reconstruction of the reflected field by adding the individual mode fields E = ∑B_lmE_lm and H = ∑B_lmH_lm. Eventually the intensity profile is determined by evaluation of

S_{z} = \frac{1}{2} (E_{r} H_{ϕ}^{*} - E_{ϕ} H_{r}^{*})

4.1. Linearly polarized regime

For the ℓ = 1 resonances, the linearly polarized regime extends from the fundamental Bragg peak up to the U(1, 2′)-cutoff (Fig. 1). In this wavelength range, coupling to ℓ = 1 modes with l > 1 does not occur because those modes do not carry a significant field within the core region. The same holds for EH₁_m modes. Up to the U (1, 2′)-cutoff, the azimuthal dependence of the HE₁_m-modes is negligible. This is evident in the observed and computed intensity rings (Fig. 5(a), 5(f), 5(b) and 5(g)) and also manifests in predominantly linear polarization. This polarization is also the cause for the strict rotation preservation of the modes. The overlap

E_{11}^{T} \cdot E_{l m}^{T *}

in the coupling integral Eq. (2) is always zero, if the modes have orthogonal polarization, meaning that the even fundamental HE₁₁ mode can only couple to even HE_1m modes (Fig. 1), no matter the form of the cross-section of the FBG. In summary, within the linearly polarized regime, all ℓ = 1 resonances are reflections into single HE_1m modes. This is also apparent through the excellent agreement between the measured and computed patterns Fig. 5(a) and 5(f) as well as 5(b) and 5(g).

Fig. 5 Linearly polarized reflections at selected resonances of the highly localized FBG, top row (a)–(e). The black arrows indicate the polarization of the reflected patterns, which coincides with launched fundamental mode. The second row ((f)–(j)) displays the computed patterns with arrows indicating the direction of the electric field. The inset (k) shows how in the case of the ℓ = 2 resonances, the linearly polarized superpositions can be constructed from equal contributions of TE, TM and HE_2m modes.

One can argue that the linearly polarized regime for the ℓ = 1 also extends to the U(1, 3′) cutoff. Strictly speaking, the polarization of the cladding modes is quasi-TE and TM already, and coupling to EH_1m is of the same order of magnitude as coupling to HE_1m modes. However, the EH_1m resonances are very close to the HE_1m (Fig. 1). Thus a superposition of both modes is observed whose polarization is predominantly linear (Fig. 5(c) and 5(h)).

For the ℓ = 2 modes the linearly polarized regime extends to the U(2, 2′) cutoff. In this regime TE, TM and HE_2m modes are degenerate. There is also an important feature arising because of their modal structure: the overlap

E_{11}^{T} \cdot E_{l m}^{T *}

exhibits a cosϕ dependence for coupling of the fundamental HE₁₁ with the TE and TM modes as well as with the HE_2m modes. In the first case, this dependence arises because a linearly polarized fundamental mode is multiplied with an azimuthally or radially polarized mode. In the second case, the cosϕ dependence results from the transverse fields of the HE_2m modes. Because of the spatial similarity of their overlaps, coupling to TE modes is as strong as coupling to even HE_2m modes and coupling to TM modes is as strong as coupling to odd HE_2m modes. The resulting superposition is linearly polarized (Fig. 5(i) and 5(j)) as observed in the experiment (Fig. 5(d) and 5(e)). The reflected patterns can best be understood by constructing four linearly polarized modes (Fig. 5(k)), analogous to the classic approximation for the 2-layer fiber in the weakly guiding limit [14

A. W. Snyder and W. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978). [CrossRef]

]: The superpositions

{LP}_{1^{'} x}^{even} = {TM}_{0 m} + {HE}_{2 m}^{even} and {LP}_{1^{'} x}^{odd} = {TE}_{0 m} + {HE}_{2 m}^{odd}

(7)

are linearly polarized along the x-axis, while

{LP}_{1^{'} y}^{even} = - {TE}_{0 m} + {HE}_{2 m}^{odd} and {LP}_{1^{'} y}^{odd} = {TM}_{0 m} - {HE}_{2 m}^{eve},

(8)

are polarized along the y-axis. Note that m is used here as a place holder for the radial index because of our choice of numbering the radial solutions (Sec. 2). Coupling to the even set of LP modes is possible as well as to odd LP modes. Thus, while the orientation of the polarization axis is conserved (Fig. 5(d), 5(e), 5(i) and 5(j)), the intensity pattern of the reflected mode can rotate. This is the case, if the modification is off axis and rotated with a certain angle to the incident polarization (Fig. 1). Box (k) in Fig. 5 shows how the two modes in Eq. (8) combine to produce the field profile in Fig. 5(i) which matches the measured result in Fig. 5(d). A similar construction holds for Fig. 5(e) and 5(j).

In conclusion we can identify a narrow regime, where all relevant cladding modes could also be described as linearly polarized LP_0′m = HE_0m and LP_1′m modes. This holds in very good approximation for cladding modes whose resonant wavelengths lie above the virtual cutoffs U(1, 3′) or U(2, 2′), which are both approximately 10 nm away from the fundamental Bragg peak (Fig. 1). Unsurprisingly, in this regime, using the scalar approach for the computation yields very good agreement [17

S. Hewlett, J. D. Love, G. Meltz, T. Bailey, and W. Morey, “Coupling characteristics of photo-induced bragg gratings in depressed-and matched-cladding fibre,” Opt. Quant. Electron. 28, 1641–1654 (1996). [CrossRef]

]. We now turn to the majority of cladding mode resonances beyond this regime, which are not linearly polarized.

4.2. Predominantly radially or azimuthally polarized regime

The quasi-TE/TM polarized regime of the spectrum starts where the double resonances appear: beyond the U(1, 3′)-cutoff. HE and EH modes are no longer degenerate; the EH resonances are shifted to the lower wavelength side. Up to the onset of l = 3 coupling at U(3, 2′), the ℓ = 1 resonances consist of single l = 1 modes. In contrast to the linearly polarized regime, the modes can be rotated in reflection, since coupling to both the even and odd set is possible.

Figure 6 displays the measured (top row, (a–d)) and computed (bottom row, (e–h)) reflected mode patterns recorded at resonances in the aforementioned wavelength regime. The azimuthal structure of the mode fields is considerably different from the EH and HE modes in the linearly polarized regime. First of all, the intensity now exhibits an azimuthal cosϕ dependence also for the ℓ = 1 resonances (Fig. 6(a) and 6(b)). Furthermore, the HE_1m mode is now predominantly azimuthally polarized (Fig. 6(a) and 6(e)). The “bow-tie” of the EH_1m (Fig. 6(b) and 6(f)) is perpendicular to the HE_1m bow-tie and exhibits a predominantly radial polarization. Thus, quasi-TE and quasi-TM polarization can be observed for l = 1 modes in this regime.

Fig. 6 Measured (top row, (a)–(d)) and computed (bottom row, (e)–(f)) mode patterns in the quasi-TE/TM regime. The patterns (a) and (c) are predominantly azimuthally polarized, while (b) and (d) exhibited radial polarization.

This observation also holds for the patterns of the ℓ = 2 resonances. The patterns displayed in Fig. 6(c) and 6(d) were predominantly azimuthally or radially polarized. In contrast to the ℓ = 1 resonances, which could be reproduced with single cladding modes, the ℓ = 2 patterns are a superposition of TE and TM modes with l = 2 hybrid HE/EH modes.

In contrast to the linearly polarized regime, the TE and TM modes are no longer degenerate after the U(0, 2′) cutoff (its definition differs from the hybrid modes and is given in A.2). The TM modes are now shifted to the shorter wavelength side and coincide with the quasi-TM polarized EH_2m modes, resulting in an almost azimuthally polarized pattern (Fig. 6 (c)). Similarly, the TE modes are degenerate with the quasi-TE polarized HE_2m modes yielding a predominately radially polarized superposition (Fig. 6 (d)). If the amplitude of the TE or TM contribution is smaller than that of the hybrid l = 2 mode, “quad-ties” emerge (Fig. 4 (c)).

4.3. Higher azimuthal order mode regime

We define a polarization as complex if modes at a resonance superpose in a way that neither yields an LP field nor a quasi-TE/TM state. Generally, such polarization states are good indicators for modes of higher azimuthal order than l = 2. In the FBG investigated, these polarizations have been observed for ℓ = 1 resonances. From the coupling constants and the cut-offs we deduced the highest possible is l = 3. Indeed the mode patterns could be reproduced by including even and odd HE_1m and HE_3m modes or EH_1m and EH_3m at the lower wavelength peak of the double resonances (Fig. 7). In contrast to the previously discussed resonances, here the four-lobed pattern does not follow cos(lϕ) anymore. Thus, the intensity pattern results unambiguously from multi-mode coupling. Indeed it can be reproduced in very good agreement using the indicated amplitudes. Note that in contrast to the other computed patterns, here the amplitudes were not derived from the computed coupling constant but are a best fit. At that point modal decomposition of the reflected cladding modes via an algorithm [18

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005). [CrossRef] [PubMed]

] or a hologram [19

T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17, 9347–9356 (2009). [CrossRef] [PubMed]

] becomes feasible because only a limited number of modes has to be considered. Such an analysis however, goes beyond the scope of this study.

Fig. 7 Reflected patterns with more complex polarization (a) and (b). The bottom row displays the superposition with the next higher degenerate azimuthal mode ((c) and (d)).

5. Conclusion and outlook

The aim of this paper was to gain a full systematic understanding of the coupling capabilities of in-fiber-components to higher order cladding modes: To what extent is the scalar approach still applicable and at which point does the full vectorial approach becomes necessary? For our endeavor, ultrafast laser written micro-void FBGs were ideal. Firstly, they allowed for very strong reflection of the fundamental mode into various cladding modes. Resulting spectra can indeed be octave-spanning [9

]. Secondly, being able to freely modify the cross-section of the FBG yields new degrees of freedom, which can be exploited to access cladding modes of higher azimuthal order.

The general coupling properties of an FBG depend on two factors. The radial distribution of its index profile determines to what azimuthal classes of modes an incident mode efficiently couples [9

]. In this regard, the fiber geometry plays a major factor, because only cladding modes with sufficient core field contributions are accessible. The azimuthal distribution with respect to the incident plane of polarization determines the orientation of the reflected mode pattern and its polarization as investigated in this paper. We could identify three principal coupling regimes: one where the cladding modes superpose to linearly polarized states, one where quasi-TE and TM modes can be observed in reflection, and a multimode regime where the polarization is more complex. These polarization regimes coincide with the calculated virtual cut-offs. Again the grating behavior is set primarily by the modal structure of the fiber rather than by the geometry of the FBG.

Our treatment and classification of the cladding modes is transferable to other fiber gratings: only Eq. (2) has to be changed to account for tilted FBGs [2

K. Lee and T. Erdogan, “Fiber mode conversion with tilted gratings in an optical fiber,” J. Opt. Soc. Am. A 18, 1176–1185 (2001). [CrossRef]

] or long period gratings [1

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997). [CrossRef]

]. It is especially of interest to evaluate the performance of fiber gratings with transversely inhomogenous cross-section, e.g. conventional UV-FBGs in large mode area fibers, femtosecond pulse induced FBGs or fusion-arc long period gratings. Prior to this work, a full treatment of such gratings seemed futile, since it was not clear what azimuthal modes had to be considered and which could be neglected. (In contrast, for transversally homogenous core gratings, cladding mode coupling is only possible to HE_1m cladding modes of the same azimuthal order and orientation as the incident fundamental mode [5

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997). [CrossRef]

].)

We anticipate the application of transversely in-homogenous fiber gratings to many fields: the cladding modes in the LP regime are especially interesting for integrated fiber lasers. In recent years, several mixed cavities have been proposed that profit from running on both core and cladding modes [20

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser & Photon. Rev. 2, 429–447 (2008). [CrossRef] [PubMed]

]. In this regard it is worth stressing that the subset of linearly polarized cladding modes conserve their polarization independent of the cross-section.

For sensing applications, one might want to harness cladding modes of higher radial order within the quasi-TE/TM regime, since they allow for interrogating the outer cladding surface. For example, the predominantly radially polarized EH modes allow for efficient excitation of surface plasmons of gold coated fibers [21

Y. Shevchenko, C. Chen, M. Dakka, and J. Albert, “Polarization-selective grating excitation of plasmons in cylindrical optical fibers,” Opt. Letters 35, 637–639 (2010). [CrossRef]

], while the strongly azimuthally polarized HE modes enable interrogation of nano particle coated fibers [22

L.-Y. Shao, J. P. Coyle, S. T. Barry, and J. Albert, “Anomalous permittivity and plasmon resonances of copper nanoparticle conformal coatings on optical fibers,” Opt. Mater. Express 1, 128–137 (2011). [CrossRef]

]. Here highly localized FBG are attractive because, like tilted FBGs, they allow for strong coupling to cladding modes of very high radial order. Elaborate schemes for directional bend sensors can be realized by accessing cladding modes of higher azimuthal order [8

A. Martinez, Y. Lai, M. Dubov, and I. Khrushchev, “Vector bending sensors based on fibre bragg gratings inscribed by infrared femtosecond laser,” Electron. Lett. 41, 472–474 (2005). [CrossRef]

Appendices

A. TE and TM modefields

In this appendix we provide all necessary expressions for the TE and TM cladding modes. The expressions for the hybrid modes are given in [9

A.1. Dispersion relations

The TE resonances were found by solving the dispersion relation

J (K p_{l} (a_{2}) + \frac{r_{l} (a_{2})}{u_{2}}) - \frac{1}{u_{2}} (K q_{l} (a_{2}) + \frac{s_{l} (a_{2})}{u_{2}}) = 0,

(9)

and the TM modes with

J (K p_{l} (a_{2}) + \frac{n_{2}^{2}}{n_{3}^{2}} \frac{r_{l} (a_{2})}{u_{2}}) - \frac{n_{2}^{2}}{n_{1}^{2}} \frac{1}{u_{2}} (K q_{l} (a_{2}) + \frac{n_{2}^{2}}{n_{3}^{2}} \frac{s_{l} (a_{2})}{u_{2}}) = 0 .

(10)

Here, we have defined

u_{l m}^{2} = {(2 π / λ)}^{2} (n_{1}^{2} - {\bar{n}}^{2}), u_{2}^{2} = {(2 π / λ)}^{2} (n_{2}^{2} - {\bar{n}}^{2}), w_{3}^{2} = {(2 π / λ)}^{2} ({\bar{n}}^{2} - n_{3}^{2}),

(11)

and

σ = i l \bar{n}, v_{21} = \frac{1}{u_{2}^{2}} - \frac{1}{u_{l m}^{2}}, v_{32} = \frac{1}{w_{3}^{2}} + \frac{1}{u_{2}^{2}}, J = \frac{J_{l}^{'} (u_{l m} a_{1})}{u_{l m} J_{l} (u_{l m} a_{1})}, K = \frac{K_{l}^{'} (w_{3} a_{2})}{w_{3} K_{l} (w_{3} a_{2})} .

(12)

The dispersion relations include Bessel-functions of the first and second kind J_n and N_n as well as modified Bessel-functions K_n of the second kind. In addition, the products

p_{l} (r) = J_{l} (u_{2} r) N_{l} (u_{2} a_{1}) - J_{l} (u_{2} a_{1}) N_{l} (u_{2} r),

(13a)

q_{l} (r) = J_{l} (u_{2} r) N_{l}^{'} (u_{2} a_{1}) - J_{l}^{'} (u_{2} a_{1}) N_{l} (u_{2} r),

(13b)

r_{l} (r) = J_{l}^{'} (u_{2} r) N_{l} (u_{2} a_{1}) - J_{l} (u_{2} a_{1}) N_{l}^{'} (u_{2} r),

(13c)

s_{l} (r) = J_{l}^{'} (u_{2} r) N_{l}^{'} (u_{2} a_{1}) - J_{l}^{'} (u_{2} a_{1}) N_{l}^{'} (u_{2} r),

(13d)

are used, where the prime stands for differentiation with respect to the total argument. Furthermore, we introduce the constant factor

C_{l m} = \frac{π a_{1} u_{l m}^{2} J_{l} (u_{l m} a_{1})}{2}

(14)

to abbreviate the following field expressions.

A.2. Virtual cut-off for TE and TM modes

Note that the dispersion relations for TE Eq. (9) and TM Eq. (10) have no discontinuities. Thus, the “virtual cut-off” definition as in [9

] does not apply in the strict sense here. We can however define a virtual cutoff U(0, 2′), where the TE and TM mode fields start to have their first ring within the core. This can be easily evaluated by setting J′₀(u₁a₁) = −J₁(u₁a₁) = 0, which happens at u₁a₁ = 3.8317, slightly before U(2, 2′).

A.3. TE and TM mode fields

In cylindrical coordinates (r, ϕ, z), the electric E and magnetic H fields of the TE cladding modes inside the core can be expressed in terms of Bessel functions J_n of the first kind. The non-zero field components in the core (r < a₁) are

E_{ϕ} = i E_{l m} u_{1} J_{0}^{'} (u_{1} r) e^{i (β z - ω t)}

(15a)

H_{z} = E_{l m} \frac{\bar{n}}{Z_{0}} \frac{u_{1}^{2}}{β} J_{0} (u_{1} r) e^{i (β z - ω t)}

(15b)

H_{r} = - i E_{l m} \frac{\bar{n}}{Z_{0}} u_{1} J_{0}^{'} (u_{1} r) e^{i (β z - ω t)},

(15c)

and inside the cladding (a₁ ≤ r ≤ a₂)

E_{ϕ}^{cl} = i E_{l m} C_{l m} u_{2} (- J r_{0} (r) + \frac{s_{0} (r)}{u_{2}}) e^{i (β z - ω t)}

(16a)

H_{z}^{cl} = E_{l m} C_{l m} \frac{\bar{n}}{Z_{0}} \frac{u_{2}^{2}}{β} (- J p_{0} (r) + \frac{q_{0} (r)}{u_{2}}) e^{i (β z - ω t)}

(16b)

H_{r}^{cl} = - i E_{l m} C_{l m} \frac{\bar{n}}{Z_{0}} u_{2} (- J r_{0} (r) + \frac{s_{0} (r)}{u_{2}}) e^{i (β z - ω t)} .

(16c)

The non-zero components of the TM modes are

E_{z} = - i E_{l m} \bar{n} \frac{u_{i}^{2}}{β} J_{0} (u_{1} r) e^{i (β z - ω t)}

(17a)

E_{r} = - E_{l m} \bar{n} u_{1} J_{0}^{'} (u_{1} r) e^{i (β z - ω t)}

(17b)

H_{ϕ} = E_{l m} \frac{1}{Z_{0}} u_{1} J_{0}^{'} (u_{1} r) e^{i (β z - ω t)}

(17c)

within the core (r < a₁) and

E_{z}^{cl} = i E_{l m} C_{l m} \bar{n} \frac{u_{2}}{β} (- J p_{0} (r) + \frac{n_{2}^{2}}{n_{1}^{2}} \frac{q_{0} (r)}{u_{2}}) e^{i (β z - ω t)}

(18a)

E_{r}^{cl} = {- \frac{u_{2}^{2} n_{1}^{2}}{u_{1} n_{2}^{2}}} E_{l m} C_{l m} \bar{n} \frac{u_{1}}{u_{2}} (- J r_{0} (r) + \frac{n_{2}^{2}}{n_{1}^{2}} \frac{s_{0} (r)}{u_{2}}) e^{i (β z - ω t)}

(18b)

H_{ϕ}^{cl} = {u_{2}} E_{l m} C_{l m} \frac{1}{Z_{0}} (- J r_{0} (r) + \frac{n_{2}^{2}}{n_{1}^{2}} \frac{s_{0} (r)}{u_{2}}) e^{i (β z - ω t)}

(18c)

inside the cladding (a₁ ≤ r ≤ a₂).

Acknowledgments

We acknowledge financial support by the German Federal Ministry of Education and Research (BMBF) under Contract No. 13N9687 and the Deutsche Forschungsgemeinschaft (Leibniz program). Jens Thomas is supported by the Carl-Zeiss-Foundation. This research was conducted with the Australian Research Council Centre of Excellence for Ultrahigh Bandwidth Devices for Optical Systems (project number CE110001018) and the assistance of the LIEF and Discovery Project programs.

References and links

1.	T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997). [CrossRef]
2.	K. Lee and T. Erdogan, “Fiber mode conversion with tilted gratings in an optical fiber,” J. Opt. Soc. Am. A 18, 1176–1185 (2001). [CrossRef]
3.	C. Lu and Y. Cui, “Fiber bragg grating spectra in multimode optical fibers,” J. Lightwave Technol. 24, 598–604 (2006). [CrossRef]
4.	D. Sáez-Rodriguez, J. L. Cruz, A. Díez, and M. V. Andrés, “Coupling between counterpropagating cladding modes in fiber Bragg gratings,” Opt. Lett. 36, 1518–1520 (2011). [CrossRef] [PubMed]
5.	T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997). [CrossRef]
6.	T. Guo, C. Chen, and J. Albert, “Non-uniform-tilt-modulated fiber bragg grating for temperature-immune micro-displacement measurement,” Meas. Sci. Technol. 20, 034007 (2009). [CrossRef]
7.	T. Guo, L. Shao, H.-Y. Tam, P. A. Krug, and J. Albert, “Tilted fiber grating accelerometer incorporating an abrupt biconical taper for cladding to core recoupling,” Opt. Express 17, 20651–20660 (2009). [CrossRef] [PubMed]
8.	A. Martinez, Y. Lai, M. Dubov, and I. Khrushchev, “Vector bending sensors based on fibre bragg gratings inscribed by infrared femtosecond laser,” Electron. Lett. 41, 472–474 (2005). [CrossRef]
9.	J. U. 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). [CrossRef] [PubMed]
10.	G. Marshall, R. 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). [CrossRef] [PubMed]
11.	R. J. Williams, C. Voigtländer, G. D. Marshall, A. Tünnermann, S. Nolte, M. J. Steel, and M. J. Withford, “Point-by-point inscription of apodized fiber bragg gratings,” Opt. Lett. 36, 2988–2990 (2011). [CrossRef] [PubMed]
12.	C. Tsao, D. Payne, and W. Gambling, “Modal characteristics of three-layered optical fiber waveguides: a modified approach,” J. Opt. Soc. Am. A 6, 555–563 (1989). [CrossRef]
13.	E. Snitzer, “Cylindrical dielectric waveguide modes,” J. Opt. Soc. Am. 51, 491–498 (1961). [CrossRef]
14.	A. W. Snyder and W. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978). [CrossRef]
15.	N. Jovanovic, J. U. 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). [CrossRef] [PubMed]
16.	B. Eggleton, P. Westbrook, C. White, C. Kerbage, R. Windeler, and G. Burdge, “Cladding-mode-resonances in air-silica microstructure optical fibers,” J. Lightwave Technol. 18, 1084–1100 (2000). [CrossRef]
17.	S. Hewlett, J. D. Love, G. Meltz, T. Bailey, and W. Morey, “Coupling characteristics of photo-induced bragg gratings in depressed-and matched-cladding fibre,” Opt. Quant. Electron. 28, 1641–1654 (1996). [CrossRef]
18.	O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005). [CrossRef] [PubMed]
19.	T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17, 9347–9356 (2009). [CrossRef] [PubMed]
20.	S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser & Photon. Rev. 2, 429–447 (2008). [CrossRef] [PubMed]
21.	Y. Shevchenko, C. Chen, M. Dakka, and J. Albert, “Polarization-selective grating excitation of plasmons in cylindrical optical fibers,” Opt. Letters 35, 637–639 (2010). [CrossRef]
22.	L.-Y. Shao, J. P. Coyle, S. T. Barry, and J. Albert, “Anomalous permittivity and plasmon resonances of copper nanoparticle conformal coatings on optical fibers,” Opt. Mater. Express 1, 128–137 (2011). [CrossRef]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.2310) Fiber optics and optical communications : Fiber optics
(060.2340) Fiber optics and optical communications : Fiber optics components
(320.7130) Ultrafast optics : Ultrafast processes in condensed matter, including semiconductors
(060.3735) Fiber optics and optical communications : Fiber Bragg gratings

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: July 18, 2012
Revised Manuscript: August 6, 2012
Manuscript Accepted: August 6, 2012
Published: September 4, 2012

Citation
Jens U. Thomas, Nemanja Jovanovic, Ria G. Krämer, Graham D. Marshall, Michael J. Withford, Andreas Tünnermann, Stefan Nolte, and Michael J. Steel, "Cladding mode coupling in highly localized fiber Bragg gratings II: complete vectorial analysis," Opt. Express 20, 21434-21449 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-19-21434

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References

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol.15, 1277–1294 (1997). [CrossRef]
K. Lee and T. Erdogan, “Fiber mode conversion with tilted gratings in an optical fiber,” J. Opt. Soc. Am. A18, 1176–1185 (2001). [CrossRef]
C. Lu and Y. Cui, “Fiber bragg grating spectra in multimode optical fibers,” J. Lightwave Technol.24, 598–604 (2006). [CrossRef]
D. Sáez-Rodriguez, J. L. Cruz, A. Díez, and M. V. Andrés, “Coupling between counterpropagating cladding modes in fiber Bragg gratings,” Opt. Lett.36, 1518–1520 (2011). [CrossRef] [PubMed]
T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A14, 1760–1773 (1997). [CrossRef]
T. Guo, C. Chen, and J. Albert, “Non-uniform-tilt-modulated fiber bragg grating for temperature-immune micro-displacement measurement,” Meas. Sci. Technol.20, 034007 (2009). [CrossRef]
T. Guo, L. Shao, H.-Y. Tam, P. A. Krug, and J. Albert, “Tilted fiber grating accelerometer incorporating an abrupt biconical taper for cladding to core recoupling,” Opt. Express17, 20651–20660 (2009). [CrossRef] [PubMed]
A. Martinez, Y. Lai, M. Dubov, and I. Khrushchev, “Vector bending sensors based on fibre bragg gratings inscribed by infrared femtosecond laser,” Electron. Lett.41, 472–474 (2005). [CrossRef]
J. U. 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. Express19, 325–341 (2011). [CrossRef] [PubMed]
G. Marshall, R. 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. Express18, 19844–19859 (2010). [CrossRef] [PubMed]
R. J. Williams, C. Voigtländer, G. D. Marshall, A. Tünnermann, S. Nolte, M. J. Steel, and M. J. Withford, “Point-by-point inscription of apodized fiber bragg gratings,” Opt. Lett.36, 2988–2990 (2011). [CrossRef] [PubMed]
C. Tsao, D. Payne, and W. Gambling, “Modal characteristics of three-layered optical fiber waveguides: a modified approach,” J. Opt. Soc. Am. A6, 555–563 (1989). [CrossRef]
E. Snitzer, “Cylindrical dielectric waveguide modes,” J. Opt. Soc. Am.51, 491–498 (1961). [CrossRef]
A. W. Snyder and W. Young, “Modes of optical waveguides,” J. Opt. Soc. Am.68, 297–309 (1978). [CrossRef]
N. Jovanovic, J. U. 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. Express17, 6082–6095 (2009). [CrossRef] [PubMed]
B. Eggleton, P. Westbrook, C. White, C. Kerbage, R. Windeler, and G. Burdge, “Cladding-mode-resonances in air-silica microstructure optical fibers,” J. Lightwave Technol.18, 1084–1100 (2000). [CrossRef]
S. Hewlett, J. D. Love, G. Meltz, T. Bailey, and W. Morey, “Coupling characteristics of photo-induced bragg gratings in depressed-and matched-cladding fibre,” Opt. Quant. Electron.28, 1641–1654 (1996). [CrossRef]
O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett.94, 143902 (2005). [CrossRef] [PubMed]
T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express17, 9347–9356 (2009). [CrossRef] [PubMed]
S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser & Photon. Rev.2, 429–447 (2008). [CrossRef] [PubMed]
Y. Shevchenko, C. Chen, M. Dakka, and J. Albert, “Polarization-selective grating excitation of plasmons in cylindrical optical fibers,” Opt. Letters35, 637–639 (2010). [CrossRef]
L.-Y. Shao, J. P. Coyle, S. T. Barry, and J. Albert, “Anomalous permittivity and plasmon resonances of copper nanoparticle conformal coatings on optical fibers,” Opt. Mater. Express1, 128–137 (2011). [CrossRef]

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