1. Introduction

A tilted fiber Bragg grating (TFBG) is a photonic structure inside which tilted periodic planes of refractive index modification are embedded into the fiber core. Tilting the grating planes with respect to the fiber axis results in strong coupling between the core mode and a multitude of back-propagating cladding modes. If the TFBG’s spectrum is observed in transmission, the coupling to cladding modes reveals itself through the production of strong resonant dips at discrete wavelengths that are shorter than the core-mode Bragg resonance wavelength [1].

The multiple sensing modalities of TFBGs, which have been widely discussed in the literature [2–4], arise from: i) differential spectral response among the cladding modes and core mode when stress is applied to the fiber and ii) inherent sensitivity of the spectral response of the cladding modes to the environment that surrounds the cladding. Apart from high resolution refractometry, surface plasmon resonance applications and multi-parameter physical sensing (strain, vibration, and temperature) [2–4], TFBGs are used as in-line fiber polarizers and broadband transmission filters [5–8].

Exposure of the fiber core to intense laser radiation is the most common method to fabricate TFBGs (and other types of FBGs). Traditionally, the permanent refractive index change is imprinted into in the doped fiber core using an interference pattern formed by two ultraviolet (UV) laser beams. The interference pattern can be produced by either amplitude/wavefront-splitting interferometers or the so-called phase mask technique. When large quantities of identical FBGs are required, the phase mask technique is favored due to its robustness and reproducibility [9].

On the other hand, the use of infrared (IR) femtosecond (fs) lasers for the inscription of different types of FBGs offers a number of advantages over the traditional UV-technology. Namely, FBG inscription with IR fs-lasers does not require i) any photosensitization of the fiber core and ii) removal of the protective polymer (e.g., acrylate or polyimide) coating and, if needed, reapplication of the coating [10–14]. Besides the fact that the latter procedure(s) is technologically involved, especially in the case of polyimide coatings, it introduces structural flaws to the fiber surface, which weakens the ultimate mechanical strength and long-term reliability of the resultant FBGs.

The inscription of TFBGs using fs-lasers has been demonstrated with different plane-by-plane techniques [15–19] and the phase mask technique [20–22]. However, the inscription of TFBGs inside coated fibers has been limited to a plane-by-plane technique so far [15]. In this respect, showing the capability of using the phase mask technique, which is generally faster, more robust, and more accurate than its plane-by-plane counterpart, for writing TFBGs inside coated fibers is highly desirable for mass production.

In this paper, we demonstrate the inscription of TFBGs using an IR fs-laser and a phase mask in non-photosensitized single-mode silica fibers through the polyimide coating. The required tilt of the Bragg grating planes inside the fiber core is achieved by simultaneously translating the fiber along its axis and the cylindrical lens (that is used to focus the fs-light along the fiber) in the orthogonal direction using piezo-actuated stages. The TFBG tilt angle is then defined by the ratio of translation ranges of the fiber and the cylindrical lens.

We also prove that the classical arrangement used for TFBG inscription with UV-lasers [2], in which it is sufficient to tilt the fiber relative to the fringe pattern, is not suitable for tight focusing geometries that are required for through-the-coating inscription [23–25]. More specifically, we show that even small rotations of the phase mask and/or the cylindrical lens in the plane of diffraction generally split the sharp line-shaped fs-laser focus and, as a consequence, dramatically decrease the peak intensity inside it by reducing the interference fringe contrast and stretching the focal volume along the beam propagation direction. The focal lines can be recombined by counter-rotating the phase mask and the cylindrical lens, but they remain normal to the interference fringes, to which the fiber axis is aligned. Therefore, tilted grating planes cannot be produced by this method inside the fiber core.

2. Background

There are mainly two approaches of fabricating TFBGs with a phase mask and UV light: i) rotating the phase mask and fiber, which also requires the rotation of the focusing cylindrical lens; ii) rotating the phase mask about the beam axis while keeping the phase mask and fiber normal to the incident beam. The inscription is usually performed in photosensitized (i.e., hydrogen/deuterium-loaded) and stripped fibers using a loosely focused UV laser beam [9].

As far as non-photosensitized fibers are concerned, FBG inscription using the phase mask technique is predominantly carried out with IR fs-lasers. However, because of the low photosensitivity of standard fibers at the often-used writing wavelength of 800 nm, the IR fs-laser beam has to be tightly focused in the fiber core through the phase mask [11]. For writing through the fiber coating, the writing beam has to be focused even tighter in order to avoid radiation damage to the fiber coating, which has a much lower damage threshold than silica glass [24]. As analyzed in [25], the confocal parameter of the laser focus inside the fiber should be less than a quarter of the fiber cladding diameter, in order to avoid damage to the protective polymer coating. For such tight focusing geometries, rotation of either the phase mask M or the cylindrical lens CL results in a focal tilt and split of the interfering diffraction orders, as illustrated in Fig. 1 in the case of rotations in the xz-plane.

 

Fig. 1. Illustration of focal tilt and split caused by rotating (a) the cylindrical lens CL only (rotation angle ξ in the xz-plane), (b) the phase mask M only (rotation angle γ in the xz-plane). M produces only 1^st diffraction order (i.e., m = ± 1). The respective diffraction angles θ₊₁ and θ₋₁ are counted from the z-axis, which coincides with the beam propagation direction. The focal splits associated with CL and M rotations are denoted by Δz_CL and Δz_M, respectively. A small angle between the split focal lines in (a) and (b) is not shown (see Appendix A).

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When both CL and M are aligned normal to the incident beam, focal lines of diffracted beams remain parallel to the phase mask, and + 1^st and –1^st orders have the same diffraction angle θ = arcsin(λ/Λ_G), where λ is the central wavelength of the incident beam, and Λ_G is the phase mask pitch. The diffraction angle θ is counted from the normal to M, which in this case coincides with the z-axis. When M remains normal to the incident beam and CL is rotated by an angle ξ (Fig. 1(a)), the diffraction angles remain the same, but the focal lines of + 1^st and –1^st orders rotate and separate in space (Δz_CL in Fig. 1(a)) along the z-axis (Appendix A and [25]). In the other limiting case, i.e., when CL remains normal to the incident beam and M is rotated by an angle γ (Fig. 1(b)), the diffraction angles (counted from the z-axis) become θ₊₁ = arcsin(λ/Λ_G – sinγ) + γ and θ₋₁ = arcsin(λ/Λ_G + sinγ) – γ, and the focal lines of + 1^st and –1^st orders also rotate and separate in space (Δz_M in Fig. 1(b)) along the z-axis. The focal lines can be recombined by counter-rotating M and CL. If the angles ξ and γ are small, it can be shown that the focal lines become merged when the condition ξ/γ ≈ tan²θ is fulfilled (see Appendix A). As an example, the ratio of the angles is ∼1.26 for a phase mask pitch Λ_G = 1.07 µm and a wavelength λ = 800 nm. The recombined focal lines are at an angle α ≈ γ/cosθ – γ with respect to the x-axis (see Appendix A).

The interference fringes are formed parallel to the bisector of + 1^st and –1^st-order diffracted beams. In the case of normal incidence, the diffracted beams are symmetric about the beam axis, and the interference fringes are thus parallel to the beam axis. However, when the phase mask is rotated, the bisector is no longer parallel to the incident beam axis. The angle β between the interference fringes and the beam axis (i.e., z-axis) is β ≈ γ/cosθ – γ under the small-angle approximation (see Appendix A). Therefore, if the focal lines are recombined (i.e., ξ/γ ≈ tan²θ), then α ≡ β (for small ξ and γ). Since the fiber axis has to be aligned parallel to the line-shaped focus, grating planes cannot be placed at a practically useful angle inside the fiber as the interference fringes are always perpendicular to the fiber axis.

Rotating the phase mask about the beam axis (i.e., z-axis), while keeping the phase mask and fiber normal to the incident beam, splits the focal lines in the xy-plane along the y-axis. In this situation, the split Δy_M is given by Δy_M = 2ztanθsinζ, where z is the distance from M to the observation xy-plane and ζ is the rotation angle about the z-axis. In the limiting case of ζ = 90°, the diffraction pattern will lie in the yz-plane and the respective focal split will be given by 2ztanθ. For tight focusing, the spot size (at the 1/e²-intensity level) along the y-axis is ≈ 2.5 µm [26], and a Δy_M = 2.5 µm, which at z = 350 µm (see experimental conditions below) corresponds to a mere ζ ≈ 0.2°, already separates the focal lines of + 1^st and –1^st orders in the xy-plane and, as a consequence, destroys the interference fringe pattern in the focal volume. This implies, as in in the case of rotations in the xz-plane, that no meaningful tilt angles for grating planes inside the fiber can be achieved for the above mask rotation geometry.

3. Experimental results

Here, we introduce minimum changes to our fs writing setup to produce tilted grating planes in the fiber core. Figure 2(a) illustrates the experimental setup for through-the-coating writing of TFBGs. Based on our recent studies on complex diffraction and dispersion effects present in the phase mask technique [27], we place the coated fiber at the optimum distance from the phase mask, i.e., where the confocal parameter of the fs-laser focus reaches its minimum. This significantly facilitates through-the-coating FBG inscription, which requires a large differential in intensity between the fiber coating and the fiber core. TFBGs are inscribed by sweeping the laser beam across the fiber core by translating CL along the y-axis and simultaneously translating the fiber along the x-axis. Since the fiber itself acts as a cylindrical lens, the refraction of the converging fs-beam at the fiber surface has to be taken into account when calculating the actual translation range of the focus inside the fiber, as shown in Fig. 2(b). This effect was not considered in [21,22], hence the reported tilt angles were grossly underestimated.

 

Fig. 2. (a) Experimental setup for writing TFBGs. (b) Refraction of the converging beam at the fiber surface (thin fiber coating is neglected).

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Let us assume that the fs-beam is initially focused at the center of the fiber core. In this case the incident rays are normal to the fiber surface. Now let us assume that the fs-beam is moved in free space (i.e., not inside the fiber) by ΔL along the y-axis. The angle of refraction $\varepsilon ^{\prime}$ in the fiber is given by Snell’s law $\sin \varepsilon ^{\prime} = \frac{{\sin \varepsilon }}{n}$, where n is the refractive index of the fiber material (silica (SiO₂) in this case) and ɛ is the angle of incidence, which is no longer 0°.

Under the third-order approximation with respect to ΔL, the transverse movement of the focus inside the fiber is then given by ΔY ≈ ΔL/n (see Appendix B). Therefore, the tilt angle of the grating planes $\phi$ in the fiber core is defined by the ratio of the fiber lateral movement ΔF and the transverse movement of the focus inside the fiber ΔY: $\phi$ = arctan(nΔF/ΔL). In theory, large angles can be achieved by changing the translation ratio of the fiber and CL. However, in practice, a finite spot size of the focus in the fiber core along the y-axis limits the maximum achievable tilt angle $\phi$. When the interference fringes with a finite width are moved obliquely in the fiber, the fringe visibility decreases as $\phi$ increases due to the partial overlap of neighbouring fringes. If the spot size in the fiber is given by 2ω_i (at the 1/e²-intensity level) along the y-axis and the intensity distribution along the x-axis follows the cos²(2πx/Λ_g)-law, the maximum achievable tilt angle will be $\phi$_max ∼ arctan(Λ_g/4ω_i).

A Ti-sapphire regenerative amplifier operating at a repetition rate of 1 kHz, central wavelength λ = 800 nm and pulse duration of 80 fs is used in the experiments. The linearly polarized fs-beam is expanded along the x-axis and focused through a zeroth-order-nulled phase mask M (Ibsen Photonics) with a pitch Λ_G = 1.07 µm using a plano-convex acylindrical lens CL with a focal length f = 15 mm, with the curved surface designed to correct spherical aberration in one dimension. The linear polarization of the fs-beam is parallel to the mask grooves. Polyimide-coated Corning SMF-28 optical fiber is placed along the line-shaped focus and aligned parallel to CL. The fiber is placed ∼350 µm away from the phase mask, where the confocal parameter of the line-shaped fs-laser focus is smallest and thus the peak intensity in the focus is highest [27]. The position of the laser line focus in the fiber core is aligned by utilizing the techniques of nonlinear photoluminescence microscopy and dark-field microscopy [26]. The piezo-actuated stages for translating CL and the fiber are triggered by the same sinusoidal function generator through a variable power splitter to control the translation ratio along the x- and y-axis. The full swing of the laser focus across the fiber core takes one second. Observations based on the dark-field microscopy technique [28] indicate that the grating fully extends across the fiber core in the z-direction.

The 15 mm length of the TFBGs is defined by the clear aperture of CL. For all the TFBGs, ΔL is kept at 15 µm (which results in ΔY ∼10 µm) to cover the entire core. In the case of pure two-beam interference pattern, the period of grating planes along the fiber axis is still given by Λ_G/2. Hence, TFBGs fabricated with different tilt angles are expected to have their main Bragg resonances positioned at the same wavelength. The TFBG transmission spectra are monitored throughout the writing process using a tunable laser and a power meter with a spectral resolution of 0.01 nm. All the TFBGs are written until saturation upon exposure to the fs-pulses at the pulse energy of 265 µJ for approximately 20 minutes. For our laser-writing conditions, this pulse energy corresponds to a peak intensity of ≈1.55×10¹³ W/cm² in the fiber core.

Figure 3 shows the transmission spectra of TFBGs written through polyimide coating with different tilt angles up to 10.3°. In each case, the cladding modes reach ∼5 dB in transmission and gradually shift to shorter wavelengths as the tilt angle increases. The cladding modes appear to have an irregular shape due to the presence of the polyimide coating, which has a higher refractive index than silica. The slight variation of the Bragg wavelength is caused by the difference in tension applied while loading the fiber into the fiber-holding jig. We note that when tilt angles are under 4°, the cladding modes are confined within a ∼20 nm wavelength range, allowing the concatenation of multiple TFBGs in a distributed sensing architecture.

 

Fig. 3. Transmission spectra of TFBGs written through the coating with different tilt angles of 4.6°, 6.8°, 8.0°, 9.0°, and 10.3°: (a) with the coating, (b) after the coating has been removed. (the spectra are offset along the vertical axis for clarity.

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Because the thickness of the polyimide coating is small (∼10 µm), TFBGs respond well to changes in the surrounding refractive index. As an example, Fig. 4 shows the transmission spectra of a TFBG with a 10.3° tilt angle in response to being immersed in liquids with different refractive indices ranged from 1.317 to 1.440 at room temperature. The accuracy of determining the cut-off wavelength is ± 0.7 nm within the spectral range of 1460–1510 nm, resulting in the surrounding refractive index sensing accuracy of 2.8 × 10⁻³. Linear regression based on the TFBG cut-off wavelength λ_c in response to the surrounding refractive index n_SRI gives a response function λ_c = 513n_SRI + 804.5, with R² = 0.9988. TFBGs with smaller tilt angles (Fig. 3(a)) are expected to preserve this response function for spectral intervals centered at shorter wavelengths.

 

Fig. 4. Transformation of the transmission spectra of a 10.3° TFBG written through the coating in response to different surrounding refractive indices.

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4. Conclusion

In conclusion, through-the-coating inscription of TFBGs using infrared femtosecond laser and the phase mask technique is demonstrated for the first time. This is achieved using a straightforward technique based on translating the focusing lens and the fiber in orthogonal directions. TFBGs with the tilt angle of up to 10.3° are fabricated with this method in non-photosensitized SMF-28 fibers coated with a ∼10 µm-thick layer of polyimide. Accurate analysis of the actual movement of the laser focus inside the fiber is provided using analytical geometry. The effectiveness of through-the-coating sensing of the surrounding refractive index is also demonstrated.

We have also proven that the classical approaches of TFBG inscription based on tilting the fiber with respect to the interference fringes cannot be used when it comes to tight focusing geometries that are required for through-the-coating inscription. Indeed, we have analytically shown that rotating either the phase mask or cylindrical lens in the plane of diffraction results in the split of the focal lines. Even though the focal lines can be recombined by counter-rotating the phase mask and cylindrical lens, no tilted grating planes can be achieved inside the fiber core due to the intrinsic orthogonality of the recombined focal line and the interference fringes. In this respect, only ‘dynamic’ laser writing, i.e., writing that requires relative translation of the laser focus and fiber, should be employed for through-the-coating inscription of TFBGs. This is in contrast to the classical approaches that represent ‘static’ laser writing as they do not require any relative translation of the laser focus and fiber.

Appendix A: Analysis of mask and lens rotation in tight focusing geometry

When both the phase mask M and focusing cylindrical lens CL are aligned normal to the incident beam, the focal lines of diffracted beams remain parallel to M, and + 1^st and –1^st orders have the same diffraction angle θ = arcsin(λ/Λ_G), where λ is the central wavelength of the incident beam, and Λ_G is the phase mask pitch. The diffraction angle θ is counted from the normal to M, which in this case coincides with the z-axis. Figure 5 illustrates the consequences of rotating either CL (Scenario 1) or M (Scenario 2).

 

Fig. 5. Illustration of focal split and tilt caused by rotating (a) the cylindrical lens CL only, (b) the phase mask M only.

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In Scenario 1, as shown in Fig. 5(a), CL is rotated by an angle ξ. Diffraction angles of + 1^st and –1^st orders remain the same, while the focal lines of + 1^st and –1^st orders rotate in the same direction as CL by ψ₊₁ and ψ₋₁, respectively. The angles ψ₊₁ and ψ₋₁ can be derived based on Fig. 5(a):

(1a)$${\psi _{ + 1}} = \arctan \left( {\frac{{\cos \theta \tan \xi }}{{1 + \sin \theta \tan \xi }}} \right),$$

(1b)$${\psi _{ - 1}} = \arctan \left( {\frac{{\cos \theta \tan \xi }}{{1 - \sin \theta \tan \xi }}} \right).$$

Then, the focal split along the z-axis is given by

(2)$$\Delta {z_{\textrm{CL}}} = z\tan \theta ({\tan {\psi_{ + 1}} + \tan {\psi_{ - 1}}} ),$$

where z is the distance from M to the midpoint between the split focal lines. For small ξ, Eq. (2) reduces to

(3)$$\Delta {z_{\textrm{CL}}} \approx 2z\xi \sin \theta .$$

In Scenario 2, as shown in Fig. 5(b), CL remains normal to the incident beam, while M is rotated by an angle γ. In this case, the diffraction angles, which are still counted from the z-axis, become θ₊₁ = arcsin(λ/Λ_G – sinγ) + γ and θ₋₁ = arcsin(λ/Λ_G + sinγ) – γ. As before, the focal lines of + 1^st and –1^st orders rotate in the same direction as M by φ₊₁ and φ₋₁, respectively. The angles φ₊₁ and φ₋₁ can be found using Fig. 5(b):

(4a)$${\varphi _{ + 1}} = \arctan \left( {\frac{{1 - \cos {\theta_{ + 1}}\tan \gamma }}{{1 + \sin {\theta_{ + 1}}\tan \gamma }}} \right),$$

(4b)$${\varphi _{ - 1}} = \arctan \left( {\frac{{1 - \cos {\theta_{ - 1}}\tan \gamma }}{{1 - \sin {\theta_{ - 1}}\tan \gamma }}} \right).$$

Then, the focal split along the z-axis is given by

(5)$$\Delta {z_\textrm{M}} \approx z({\tan {\varphi_{ - 1}} - \tan {\varphi_{ + 1}}} )/\cos \theta $$

For small γ, Eq. (5) reduces to

(6)$$\Delta {z_\textrm{M}} \approx 2z\gamma {\sin ^2}\theta \tan \theta .$$

Equations (3) and (6) show that for our 1^st-order mask the focal splits will be Δz_CL ≈ 9.1 µm µm and Δz_M ≈ 7.7 µm per one degree of rotation at z = 350 µm, i.e., where the fiber was placed during the inscription. Taking into account that the confocal parameter of our beam in free space is ≈11 µm at the optimum position z = 350 µm [27], focal splitting caused by introducing ξ or γ at the level of 1° will become comparable with the confocal parameter and, as a result, the interference fringe contrast in the focal volume will be reduced dramatically. In order to have the focal lines overlapped in space at a distance z from M, the angles ξ and γ have to satisfy the condition Δz_CL = Δz_M (Eqs. (3) and (6)). For small ξ and γ, the above condition is met when ξ/γ ≈ tan²θ.

Now we show that the focal lines after M can be merged by counter-rotating M and CL, as shown in Fig. 6, with the angle between M and the merged focal lines being denoted by χ.

 

Fig. 6. Merging the focal lines by counter-rotating the phase mask M and cylindrical lens CL.

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The angle χ can be found using the following considerations. The angles shown in Fig. 6 are related to each other as

(7)$$\tan \chi = \frac{{\sin ({\xi + \gamma } )\cos ({{\theta_{ + 1}} - \gamma } )}}{{\cos \xi + \sin ({\xi + \gamma } )\sin ({{\theta_{ + 1}} - \gamma } )}} = \frac{{\sin ({\xi + \gamma } )\cos ({{\theta_{ - 1}} + \gamma } )}}{{\cos \xi - \sin ({\xi + \gamma } )\sin ({{\theta_{ - 1}} + \gamma } )}}.$$

By applying the small-angle approximation to Eq. (7), one arrives at the same relationship for ξ and γ, namely ξ/γ ≈ tan²θ. Finally, for small angles, χ can be written as

(8)$$\chi \approx {\gamma \mathord{\left/ {\vphantom {\gamma {\cos \theta }}} \right.} {\cos \theta }}.$$

Therefore, the angle α by which the focal lines are rotated with respect to the x-axis is

(9)$$\alpha = \chi - \gamma = {\gamma \mathord{\left/ {\vphantom {\gamma {\cos \theta - \gamma }}} \right.} {\cos \theta - \gamma }}.$$

When the phase mask is rotated by γ, the angle β between the interference fringes and the z-axis is given by

(10)$$\beta = \frac{1}{2}({{\theta_{ - 1}} - {\theta_{ + 1}}} )\approx {\gamma \mathord{\left/ {\vphantom {\gamma {\cos \theta }}} \right.} {\cos \theta }} - \gamma .$$

As one can see, α ≡ β for small angles. Therefore, the interference fringes always remain normal to the focal line, to which the fiber is aligned.

Appendix B: Analysis of the laser focus movement inside the fiber

Figure 7 shows the geometry of focusing a fs-beam inside the fiber, with the focusing half-angle being denoted by η. Let us assume that the fs-beam is initially focused at the center of the fiber core, which is represented by the (0, ΔL)-point. In this case, the incident rays are normal to the fiber surface. The fs-beam is then moved in free space by ΔL along the y-axis. The incident rays a and b are intercepted by the fiber surface at the (z₁, y₁)-point and (z₂, y₂)-point, respectively. Because of the transverse shift ΔL, the rays a and b are now focused at the (z₀, y₀)-point inside the fiber. The angles of incidence and refraction are denoted by ${\varepsilon _1}$ and ${\varepsilon _2}$ and ${\varepsilon ^{\prime}_1}$ and ${\varepsilon ^{\prime}_2}$, respectively.

 

Fig. 7. Illustration of refraction of the converging fs-IR beam at the fiber surface (thin fiber coating is not shown).

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As the fiber surface is represented by a circle of radius R centered at (0, ΔL), the coordinates of the intercept points (z₁, y₁) and (z₂, y₂) can be found from the circle equation and the slopes of the rays a and b:

(11a)$${z_1} = \frac{{ - \Delta L\tan \eta - \sqrt {{R^2} + {R^2}{{\tan }^2}\eta - \Delta {L^2}} }}{{1 + {{\tan }^2}\eta }},\,\,{y_1} ={-} {z_1}\tan \eta ,$$

(11b)$${z_2} = \frac{{\Delta L\tan \eta - \sqrt {{R^2} + {R^2}{{\tan }^2}\eta - \Delta {L^2}} }}{{1 + {{\tan }^2}\eta }},\,\,{y_2} = {z_2}\tan \eta .$$

The angles of incidence ɛ₁ and ɛ₂ are defined with respect to the radial lines that go through (z₁, y₁) and (z₂, y₂). The slopes of the radial lines are given by

(12a)$$\frac{{{y_1} - \Delta L}}{{{z_1}}} = \frac{{ - \tan \eta \sqrt {{R^2} + {R^2}{{\tan }^2}\eta - \Delta {L^2}} + \Delta L}}{{\Delta L\tan \eta + \sqrt {{R^2} + {R^2}{{\tan }^2}\eta - \Delta {L^2}} }},$$

(12b)$$\frac{{{y_2} - \Delta L}}{{{z_2}}} = \frac{{ - \tan \eta \sqrt {{R^2} + {R^2}{{\tan }^2}\eta - \Delta {L^2}} - \Delta L}}{{\Delta L\tan \eta - \sqrt {{R^2} + {R^2}{{\tan }^2}\eta - \Delta {L^2}} }}.$$

The angles of incidence ɛ₁ and ɛ₂ then become

(13)$$\tan {\varepsilon _1} = \tan {\varepsilon _2} = \frac{{\Delta L}}{{\sqrt {{R^2} + {R^2}{{\tan }^2}\eta - \Delta {L^2}} }}.$$

The angles of refraction can be found from Snell’s law and Eq. (13):

(14)$$\tan {\varepsilon ^{\prime}_1} = \tan {\varepsilon ^{\prime}_2} = \frac{{\Delta L}}{{\sqrt {{n^2}{R^2} + {n^2}{R^2}{{\tan }^2}\eta - \Delta {L^2}} }},$$

where n is the refractive index of the fiber material. The angles between the refracted rays and the z-axis are $\pi - \eta + {\varepsilon _1} - {\varepsilon ^{\prime}_1}$ and $\eta + {\varepsilon _2} - {\varepsilon ^{\prime}_2}$, respectively. The slopes of the refracted rays are given by quite complicated relationships, which are not shown here. However, if only linear terms of ΔL are retained, the slopes of the refracted rays a and b respectively reduce to

(15a)$$\frac{{{y_1} - {y_0}}}{{{z_1} - {z_0}}} \approx \left( {1 - \frac{1}{n}} \right)\frac{{\Delta L}}{{R\cos \eta }} - \tan \eta ,$$

(15b)$$\frac{{{y_2} - {y_0}}}{{{z_2} - {z_0}}} \approx \left( {1 - \frac{1}{n}} \right)\frac{{\Delta L}}{{R\cos \eta }} + \tan \eta .$$

By knowing the slopes of the refracted rays and the coordinates of points lying on them, one can find the intersection point of the refracted rays, i.e., the focal point inside the fiber. In the linear approximation with respect to ΔL, the coordinates of the focal point are given by

(16a)$${z_0} \approx 0,$$

(16b)$${y_0} \approx \frac{{({n - 1} )}}{n}\Delta L.$$

In the quadratic approximation with respect to ΔL, the coordinates of the focal point are

(16c)$${z_0} \approx \frac{{({n - 1} )\Delta {L^2}\cos \eta }}{{{n^2}R}},$$

(16d)$${y_0} \approx \frac{{({n - 1} )}}{n}\Delta L.$$

Finally, the transverse movement of the focus inside the fiber is

(17)$$\Delta Y = \Delta L - {y_0} \approx \frac{{\Delta L}}{n},$$

with a high degree of accuracy.

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