Opto-mechanical fiber sensing with optical and acoustic cladding modes

Zadok, Avi; Zehavi, Elad; Bernstein, Alon

doi:10.1063/5.0147301

Optical fibers are an excellent sensor platform. However, the detection and analysis of media outside the cladding and coating of standard fibers represent a long-standing challenge: light that is guided in the single optical core mode does not reach these media. Cladding modes help work around this difficulty, as their transverse profiles span the entire cross-section of the fiber cladding and reach its outer boundary. In this tutorial, we introduce and discuss in detail two recent advances in optical fiber sensors that make use of cladding modes. Both concepts share optomechanics as a common underlying theme. First, we describe a spatially continuous distributed analysis using the optical cladding modes of the fiber. Light is coupled to these modes using Brillouin dynamic gratings, which are index perturbations associated with acoustic waves in the core that are stimulated by light. Unlike permanent gratings, which are routinely used to couple light with cladding modes, Brillouin dynamic gratings may be switched on and off at will and can be confined to short fiber sections at arbitrary locations in a random-access manner. Second, we present the extension of the cladding mode sensor concept to include acoustic rather than optical modes. The acoustic cladding modes may be stimulated and monitored by guided light in the single core mode, and their linewidths are modified by the elastic properties of surrounding media. The principles and analyses of both concepts are provided in detail, alongside examples of experimental setups and results.

I. INTRODUCTION

Standard optical fibers constitute an exceptional sensing platform.^1–4 Their low cost and low propagation losses allow for installation over long reaches and for remote measurements from a long stand-off distance. Their small cross-section permits their installation within many structures with little disruption of function. Fibers are comparatively immune to electromagnetic interference, and they are suitable for installation in harsh and hazardous environments where the use of electricity might be prohibited. In addition, optical fibers support spatially continuous distributed analysis, in which each fiber section becomes an independent node in a sensor network.⁵ Optical fiber sensors have, therefore, been studied and employed for almost fifty years.

Most sensors over standard, single-mode optical fibers observe the effect of a quantity of interest on the light that is guided in the core of the fiber. Examples include the monitoring of temperature, static strain, sound and vibrations, electrical and magnetic fields, gamma radiation, and more.^1–5 The condition being observed may alter the magnitude, frequency, phase, or polarization of guided light, and those changes in turn may vary with the optical frequency.^1–5 Optical fiber sensors progress rapidly and become increasingly sensitive, sophisticated, and useful. However, the fundamental sensing configuration over single-mode fibers faces a significant limitation: the condition being observed must prevail in spatial overlap with the inner core. While this requirement is more easily fulfilled in the measurements of temperature and axial strain, for example, it restricts the detection and sensing of media outside the boundaries of the fiber cladding and coating. The optical fiber sensing of chemical reagents and biological species represents an inherent, long-standing difficulty.

Over the last decades, a large number of clever, sensitive, and target-specific optical fiber sensors for the analysis of outside media have nevertheless been proposed, demonstrated, and successfully employed. For an excellent review, see Ref. 6. The sensing concepts may be broadly classified into several categories. One useful approach is that of indirect sensing, in which a transduction mechanism converts the presence or concentration of a target reagent to a secondary signal that may be picked up by light in the core of the fiber, most often a change in temperature or strain.⁶ Although the sensors directly measure modifications induced within the core, they may indirectly identify conditions prevailing outside the cladding or coating. An example of indirect sensing is the monitoring of hydrogen through strain induced by the swelling of palladium coating layers.⁷ In many cases, the implementation of the transduction mechanism requires local modifications, and it is difficult to scale to long distances and spatially distributed analysis.

A second category relies on structural modifications to the standard fiber geometry that bring the guided light into direct spatial overlap with the medium under test. Modifications may take the form of etching, tapering, cleaving of facets, the formation of through holes, and more.^6–14 This strategy can be very effective. However, similar to most indirect approaches, it is difficult to scale beyond point-measurements at a few discrete and pre-selected locations. A third approach is based on moving away from the standard single-mode fiber altogether. Instead, sensors are implemented in non-standard cross-sections, such as photonic crystals and nano-structured fibers,^15–19 in which test liquids and gases may flow through holes. Alternatively, the fiber can be made of reactive, biologically sensitive materials instead of silica.^20–23 The lengths of non-standard fibers are sometimes limited, and their availability to the broader community can be restricted as well.

All sensing approaches described earlier rely on the propagation of light in the fundamental, single optical core-mode supported by the fiber. However, the transverse profiles of standard fibers also support a large category of so-called cladding modes. These include optical cladding modes that guide light^24–29 as well as mechanical (acoustic) cladding modes for the propagation of elastic perturbations.^30–33 Unlike the single optical core mode, the transverse profiles of both optical and acoustic cladding modes span the entire cross-section of the fiber cladding and reach its outer boundaries. Therefore, the propagation of optical and/or acoustic waves in the cladding modes of the fiber provides unique opportunities for the sensing of outside media beyond those available through the optical core mode.

Of the two categories of cladding modes, the optical ones are much better known in the fiber-optics community. While these modes represent a distraction from a telecommunication standpoint, their potential in fiber sensing applications is widely recognized and employed. The transverse profiles of the optical cladding modes include evanescent tails, which may overlap with the media under test. Therefore, the coupling of light to the cladding modes and the propagation in these modes may be affected by the properties of outside substances. Cladding-mode-based sensors have been used in thousands of works over the past thirty years.^24,26–28

The main challenge associated with optical cladding mode sensors revolves around the coupling of light to and from the single optical core mode. In most cases, light is launched and collected at the ends of the fiber in the single core mode only. Coupling to the cladding modes takes place along the fiber through the inscription of permanent gratings in the core.^{24,25,29,34,35} Short period gratings can couple light between a core mode and a counter-propagating cladding mode.^24,25,29 Long-period gratings can do the same between co-propagating core and cladding modes.^25,29 However, gratings are implemented in discrete, specific locations. As a result, cladding-mode sensors that are based on grating interfaces provide point measurements only. The sensors can be cascaded to form quasi-distributed networks; however, their extension toward spatially continuous distributed analysis is inherently difficult.

The acoustic cladding modes of standard optical fibers have been analyzed and characterized experimentally for the first time in 1985.³⁰ Like their optical counterparts, the transverse profiles of acoustic cladding modes reach the outer boundary of the cladding, and their propagation may be affected by outside conditions. Compared with optical cladding modes, optical fiber sensing through guided acoustic modes is difficult to carry out directly: acoustic waves in fibers are seldom launched or detected in the mechanical domain, as interfaces at the ends of the fibers are optical. Instead, acoustic waves in fibers can be stimulated by guided optical waves through electrostriction and may be monitored based on photoelastic scattering of light.^{30–33,36,37}

The combination of electrostriction and/or photoelasticity, often referred to as Brillouin scattering in fibers,^36,37 is highly useful for the indirect sensing of parameters of interest based on their effect on acoustic waves, which serve as mediators. This prospect is best realized in backward Brillouin scattering optical fiber sensors, which are successfully commercialized for spatially distributed measurements of temperature and strain.^5,38 However, the acoustic modes involved in traditional backward Brillouin scattering sensors are guided by the core of the fiber, similar to the single optical core mode. These sensors are, therefore, not suitable for the analysis of media outside the cladding. The potential of Brillouin scattering processes involving acoustic cladding modes toward sensing applications has not been considered until recently.

This tutorial is intended to introduce and describe two recent developments in the field of optical fiber sensing, which share the use of guided cladding modes and opto-mechanical interactions as common themes. One concept is based on the coupling spectra of light between the optical core mode and an optical cladding mode.^39,40 Unlike established sensing protocols based on optical cladding modes, the coupling of light to these modes does not rely on the inscription of permanent gratings. Instead, the coupling is realized by photoelastic perturbations that are induced through a backward Brillouin scattering process in the core mode.^41,42 The second concept involves the guided acoustic cladding modes of the fiber.^43–54 These modes are stimulated and monitored by guided light in the core mode, and their oscillations indirectly convey information regarding conditions outside the cladding.^43–54

In contrast to most traditional strategies for sensing chemicals using optical fibers, the two concepts are realized over standard, unmodified single-mode fibers. Both methods are suitable for spatially continuous distributed analysis, and both also support operation in fibers with certain types of coatings. Brought together, the new sensors allow for monitoring the optical and elastic properties of media where light in the core mode does not reach, thereby addressing long-standing challenges. Current performance, in terms of the number of resolution points, measurement range, sensitivity, and specificity, remains modest. Yet the measurement protocols hold much promise for further progress. We hope the following will help researchers and engineers working in fiber optics become familiar with these developments.

II. CLADDING MODES OF STANDARD OPTICAL FIBERS

The cross-section of standard optical fibers used in cladding mode sensing experiments consists of three distinct regions: an inner core of few-micrometers radius made of germanium-doped silica, a pure silica cladding with a standard diameter of 125 µm, and a protective polymer coating with thickness between a few micrometers and tens of micrometers [see Fig. 1(a)]. The fiber is placed in a medium under test, which is assumed to extend infinitely. Cladding modes are guided along the fiber axis due to the index contrast between the cladding and coating and the surrounding environment. Guiding requires that the refractive index of the coating n_coat be lower than that of pure silica, n_SiO2 = 1.444 refractive index units (RIU) at 1550 nm wavelength, and that optical absorption in the coating not be exceedingly high within one resolution cell.

FIG. 1.

$(a): Illustration of the transverse cross-section of a standard, coated step-index fiber in an infinite external medium. (b) Illustration of the three-layer model used in analysis. The medium outside the cladding is taken to be infinite and uniform (see text). The refractive indices of the core, cladding, coating, and outside medium are noted by ncore, nSiO2, ncoat, and next, respectively.$

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(a): Illustration of the transverse cross-section of a standard, coated step-index fiber in an infinite external medium. (b) Illustration of the three-layer model used in analysis. The medium outside the cladding is taken to be infinite and uniform (see text). The refractive indices of the core, cladding, coating, and outside medium are noted by n_core, n_SiO2, n_coat, and n_ext, respectively.

Unfortunately, many of the standard coating layers, such as the most common acrylate layers, are characterized by n_coat > n_SiO2. This condition is beneficial for telecommunications and is, therefore, the norm in commercial fiber. Such coating layers must be removed for cladding modes to be guided. Certain types of polymers were developed to serve as secondary claddings with a refractive index below that of silica.⁵⁵ Fibers coated with these polymers support the propagation of cladding modes and may be used for sensing purposes with their coating intact. Examples shown later in this tutorial make use of fluoroacrylate coating layers with an n_coat of 1.405 RIU.⁵⁵ It is assumed below that the refractive index n_ext of the medium outside the coating is lower than n_coat.

The analysis and modeling of cladding mode sensors are based on calculations of the normalized transverse profiles and effective indices of the cladding modes. In the most general case, the electromagnetic wave equation needs to be solved in four distinct domains: core, cladding, coating, and outside medium, and the boundary conditions at the three interfaces between adjacent domains must be satisfied. Yet, simplifying assumptions will be made for the purpose of this discussion. As explained below, these assumptions do not compromise the validity of the analysis for cases of practical interest. The reader is referred to earlier literature for a more complete analysis.²⁵

Let us consider the functional form of the optical field transverse profile within the coating layer. If the effective index of a cladding mode n_eff is higher than n_coat, the modal profile decays exponentially within the coating. Decay takes place typically at sub-micron depths. Therefore, the evanescent tails of this category of modes do not reach into the surrounding medium, even for the thinnest of polymer coatings. If instead n_eff < n_coat, the cladding mode profile is oscillating across the coating layer and reaches into the outside substance. In principle, this latter category of optical cladding modes could have a large advantage in chemical sensing.

However, the contrast in refractive indices between n_SiO2 and n_coat is comparatively large: about 0.04 RIU in the above-mentioned example. Consequently, the order m of cladding modes for which n_eff < n_coat is high: several tens or even higher. The transverse profiles of these very high-order modes oscillate rapidly across the silica cladding and even within the small inner core. Due to these rapid radial oscillations, the spatial overlap between the transverse profiles of these high-order cladding modes and the single core mode cancels out almost entirely. As will be discussed in Sec. III, such overlap is essential for the coupling of light between core and cladding modes. Therefore, in most practical scenarios with coated fibers, the cladding modes that reach outside the coating remain largely inaccessible. We, therefore, restrict the analysis to cladding modes for which n_eff > n_coat.

Referring again to the fluoroacrylate coating, the cladding modes with the largest spatial overlap with the core mode decay within less than a micron of the coating. We may, therefore, consider that layer to be infinitely wide and disregard the medium beyond the coating [Fig. 1(b)]. Note that cladding mode sensing of chemicals with coated fibers is still possible, even if indirectly, based on modifications to n_coat induced by target reagents (see Sec. IV). Direct sensing of media outside the coating would require coating layers with an index much closer to n_SiO2. In what follows, we consider an infinite and uniform medium outside the silica fiber cladding, with index n_ext or n_coat for bare or coated fibers, respectively. For simplicity, we refer below to the index outside the cladding as n_ext for both cases.

We restrict our discussions to cladding modes with transverse profiles that are radially symmetric and independent of the transverse azimuthal coordinate. While cladding modes with azimuthal symmetries of any discrete order are supported by the fiber, their spatial overlap integrals with the radially symmetric single core mode and/or the transverse profiles of radially symmetric grating perturbations vanish. The coupling of light to cladding modes of general azimuthal symmetry is, therefore, more difficult, and they are disregarded in the subsequent treatment.

The boundary conditions' equation for the optical cladding modes is given in Appendix A. Its numerical solutions yield the effective indices

n_{eff}^{(m)}

and transverse profiles

{\vec{e}}^{(m)} (r)

(m⁻¹) of a discrete set of radially symmetric modes. Here, m is an integer modal order. The transverse profiles are normalized so that

2 π \int_{0}^{\infty} n (r) {\vec{e}}^{(m)}^{†} (r) {\vec{e}}^{(m)} (r) r d r = 1

⁠. Here,

n (r)

is the local refractive index, which differs between the core, cladding, and external medium, as mentioned earlier. Figure 2(a) shows the normalized transverse profile of the electro-magnetic intensity for mode m = 5. A step-index core with index n_core = 1.449 RIU and radius a_core = 4.1 µm was assumed. The cladding radius a_clad was 62.5 µm. Figure 2(b) presents the effective index as a function of modal order m for n_ext = 1.405 RIU. Figure 2(c) shows the relative fraction of electromagnetic intensity in overlap with the outer medium,

σ^{(m)} \equiv 2 π \int_{a_{clad}}^{\infty} n (r) {\vec{e}}^{(m)}^{†} (r) {\vec{e}}^{(m)} (r) r d r .

(1)

The fraction

σ^{(m)}

increases with modal order. The higher-order cladding modes are expected to support more sensitive measurements of changes to n_ext. However, the coupling of light from the core mode is degraded for high-order modes due to spatial overlap considerations, as noted earlier.

FIG. 2.

$(a) Calculated normalized transverse profile of electromagnetic intensity in a cladding mode of order m = 5, in a step-index, single-core-mode fiber. The radii of the core and cladding were 4.1 and 62.5 µm, respectively. The indices of the core, cladding, and surrounding medium were 1.449, 1.444, and 1.405 RIU, respectively. (b) Calculated effective indices of the cladding modes guided by the same fiber as a function of modal order. (c) Calculated relative fraction of the electromagnetic intensity in overlap with the outside medium for the cladding modes of the same fiber as a function of modal order.$

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(a) Calculated normalized transverse profile of electromagnetic intensity in a cladding mode of order m = 5, in a step-index, single-core-mode fiber. The radii of the core and cladding were 4.1 and 62.5 µm, respectively. The indices of the core, cladding, and surrounding medium were 1.449, 1.444, and 1.405 RIU, respectively. (b) Calculated effective indices of the cladding modes guided by the same fiber as a function of modal order. (c) Calculated relative fraction of the electromagnetic intensity in overlap with the outside medium for the cladding modes of the same fiber as a function of modal order.

Figure 3(a) shows the calculated variation in the effective index $n_{eff}^{(15)}$ as a function of changes to n_ext about a baseline value of 1.405 RIU. The changes in $n_{eff}^{(15)}$ are smaller than those in n_ext by a factor of about 1000 due to the small fraction of the modal intensity in overlap with the outside medium. The small changes in $n_{eff}^{(m)}$ are nevertheless measurable. Possible variations in the exact value of a_clad may introduce ambiguity in data analysis. Figure 3(b) presents the changes in the same effective index $n_{eff}^{(15)}$ as a function of a_clad. Radius variations of 0.1 µm modify the effective index by about 10⁻⁵ RIU.

FIG. 3.

$(a) Changes to the effective index of a cladding mode of order m = 15 as a function of variations in the refractive index of the outside medium. For the fiber parameters, see Fig. 2. The dashed red line denotes the index of fluoroacrylate coating used in this work. (b) Changes to the same effective index as a function of variations in the cladding radius. The dashed red line denotes the nominal radius of 62.5 µm.$

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(a) Changes to the effective index of a cladding mode of order m = 15 as a function of variations in the refractive index of the outside medium. For the fiber parameters, see Fig. 2. The dashed red line denotes the index of fluoroacrylate coating used in this work. (b) Changes to the same effective index as a function of variations in the cladding radius. The dashed red line denotes the nominal radius of 62.5 µm.

III. COUPLING OF LIGHT FROM THE CORE MODE TO A CLADDING MODE USING BRILLOUIN DYNAMIC GRATINGS

As an alternative to the inscription of permanent gratings, coupling between the core and cladding modes can be based on so-called Brillouin dynamic gratings: the traveling photoelastic perturbations in the refractive index associated with backward stimulated Brillouin scattering.^36,37 To introduce this concept, let us first consider the coupling of light between counter-propagating optical core and cladding modes using standard, permanent fiber Bragg gratings. Figure 4(a) illustrates the dispersion relations between the temporal angular frequencies ω and axial wavenumbers $β = k_{0} n_{eff} (ω)$ of optical waves in the fiber. Here $k_{0}$ is the vacuum wavenumber. Each point represents a valid solution to the optical wave equation. The points connect to form continuous traces, corresponding to distinct spatial guided modes. The exact dependence of $ω (β)$ in each mode can become rather complex. For the sake of this discussion, however, we may well approximate the dispersion relation of a given optical mode as a straight line with a slope that equals its phase velocity: c/n_eff. Here, c is the speed of light in a vacuum.

FIG. 4.

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(a) Dispersion relations between axial wavenumber and angular optical frequency in the optical core mode in one direction and an optical cladding mode in the opposite direction. The two are characterized by straight lines in the dispersion map, with slopes given by their respective phase velocities. Light is coupled between the two modes by a fixed Bragg grating, represented by a horizontal line corresponding to its wavenumber contribution. (b) Illustration of backward Brillouin stimulation of a guided acoustic core mode by two optical pump waves, which propagate in the single optical core mode in opposite directions. The two pumps are spectrally detuned by the Brillouin frequency shift of the fiber. With that choice, the frequency and wavenumber of the electrostrictive force induced by the two pump waves match those of a dilatational, axial acoustic core mode with phase velocity v_L. (c) Photoelastic scattering of a probe wave from the optical core mode to a counter-propagating cladding mode by the acoustic wave stimulated by the two pumps. The frequency and wavenumber of the nonlinear polarization term due to photoelasticity match those of an optical cladding mode for a specific choice of the probe optical frequency.

Two modes are shown in Fig. 4(a): the single core mode propagating in the positive axial direction

\hat{z}

⁠, and a counter-propagating cladding mode. The wavenumbers of the two modes are, therefore, positive and negative, respectively. Consider two fields of the same angular frequency ω, one in each of the modes shown. Their wavenumbers differ by

Δ k = \frac{ω (n_{eff}^{(c o r e)} + n_{eff}^{(m)})}{c} .

(2)

Here,

n_{eff}^{(c o r e)}

is the effective index of the single core mode, and

n_{eff}^{(m)}

denotes the effective index of a specific cladding mode, order m. The coupling of light between the modes requires compensation for the wavenumber mismatch Δk.²⁵ The wavenumbers may be matched using an axial grating of refractive index perturbations, with a period Λ determined by the Bragg condition,²⁵

\frac{2 π}{Λ} = Δ k = \frac{ω (n_{eff}^{(c o r e)} + n_{eff}^{(m)})}{c} .

(3)

The grating is illustrated in the dispersion relation of Fig. 4(a) as a horizontal line of length Δk. Since the grating is static, it is not associated with an angular frequency of propagation. Light may be scattered between the counter-propagating core and cladding modes with no change in optical frequency. Equation highlights the potential of coupling to the cladding modes toward sensing applications. Since

n_{eff}^{(m)}

varies with the index of the surrounding medium n_ext [see Fig. 3(a)], the coupling spectrum induced by a fixed grating is sensitive to an outside substance under test. While static gratings support thousands of demonstrations of optical fiber cladding mode sensors, their operation is restricted to the points of permanent inscription.

A. Stimulation of Brillouin dynamic gratings by optical pump waves

The propagation of acoustic waves in a medium is associated with moving perturbations to its refractive index by virtue of photoelasticity. Therefore, the stimulation of acoustic waves with the necessary wavenumber Δk could, in principle, replace permanent inscription in the coupling of guided light between spatial modes. Such stimulation is possible through the widely known and employed process of backward Brillouin scattering in fibers.^41,42 The dispersion relations of the waves involved in the process are illustrated in Fig. 4(b). A first optical pump wave of angular frequency ω_p1 propagates in the core mode in the positive $\hat{z}$ direction. We denote the transverse profile of the optical field in the core mode as ${\vec{e}}^{(c o r e)} (r)$ (m⁻¹) and the magnitude of the first pump wave in volts as A_p1. A second pump wave is counter-propagated in the core mode in the negative $- \hat{z}$ direction, with magnitude A_p2 and optical angular frequency ω_p2 = ω_p1 − Ω.

The angular frequency offset Ω is chosen near the Brillouin frequency shift of the fiber, Ω_B (see further below). We assume, for the time being, that both pump waves are continuous with fixed magnitudes. We also assume for simplicity that the two waves are co-polarized along the entire fiber (for vector analysis of backward stimulated Brillouin scattering in standard, weakly birefringent fibers, see Ref. 56; for mitigation of polarization-induced fading, see Refs. 38 and 57).

The beating between the two pump waves induces an electrostrictive force per unit volume in the fiber medium, which is almost entirely aligned with the axial direction

\hat{z}

⁠.^36,37 It is given by

\begin{align} \vec{F} (r, z, t) & = j \frac{q a_{2}}{4 n_{eff}^{(c o r e)} c} P_{12} (Ω) [{\vec{e}}^{(c o r e) †} (r) {\vec{e}}^{(c o r e)} (r)] \\ \times \exp (j q z - j Ω t) \hat{z} + c . c . \end{align}

(4)

In Eq. , z is the axial coordinate, t stands for time, a₂ = −1.19 is a photoelastic constant of silica,³² and

P_{12} (Ω)

(W) is the beating power between the two pump waves,

P_{12} (Ω) = 2 n_{eff}^{(c o r e)} ε_{0} c A_{p 1} A_{p 2}^{*} .

(5)

In Eq. , ɛ₀ is the vacuum permittivity. The transverse profile of the force per unit volume

\vec{F}

is confined to the core of the fiber. The force per unit volume propagates in the positive

\hat{z}

direction, with an angular frequency Ω and a wavenumber that equals

q = \frac{n_{eff}^{(c o r e)} (ω_{p 1} + ω_{p 2})}{c} \approx 2 \frac{n_{eff}^{(c o r e)} ω_{p 1}}{c} .

(6)

The force per unit volume is shown in the dispersion relation [Fig. 4(b)] alongside the two pump waves. The force may stimulate the oscillations of acoustic waves that are guided by the fiber. Stimulation requires that the acoustic wave match the properties of the driving force: both the material displacement vector and the wave-vector should point in the axial direction. Therefore, acoustic waves that are relevant to backward stimulated Brillouin scattering are dilatational. The velocity of dilatational acoustic waves in silica v_L equals 5996 m × s⁻¹. In addition, the transverse profile of the acoustic mode should be largely confined to the core, similar to the single optical core mode, for efficient spatial overlap.

Step-index, standard single-mode fiber supports dilatational acoustic modes that propagate in the axial direction with large confinement to the core. We denote the transverse profile of axial material displacement in one such acoustic core mode as

{\vec{e}}_{z}^{(a c)} (r)

(m⁻¹). The transverse profile is normalized so that

2 π \int_{0}^{\infty} {|{\vec{e}}_{z}^{(a c)} (r)|}^{2} r d r = 1

⁠. Solutions for the transverse profiles are discussed in Appendix B. While the displacement vectors also include nonzero radial components, they are typically much weaker than the axial ones, and they are neglected below. Figure 4(b) also shows the dispersion relation of one such mode. It is well approximated by a straight line of slope v_L. Stimulation of the acoustic mode is most effective when the electrostrictive force per volume matches its angular frequency and wavenumber [Fig. 4(b)]. This condition is met for a specific angular frequency detuning between the two pump waves,

Ω = q v_{L} = 2 n_{eff}^{(c o r e)} ω_{p 1} \frac{v_{L}}{c} \equiv Ω_{B} .

(7)

For standard fibers and at the telecommunication wavelength of 1550 nm, Ω_B equals approximately 2π × 10.85 GHz.^36–38

The displacement (m) of the stimulated acoustic mode is given by

\begin{align} {\vec{U}}^{(a c)} (r, z, t) & = b^{(a c)} (Ω) {\vec{e}}_{z}^{(a c)} (r) \exp (j q z - j Ω t) \hat{z} + c . c . \\ = U_{z}^{(a c)} \hat{z} + c . c, \end{align}

(8)

with a modal magnitude

b^{(a c)}

(m²)^36,37

b^{(a c)} (Ω) = j \frac{q a_{2}}{4 n_{eff}^{(c o r e)} c ρ_{0}} Q_{E S}^{(c o r e)} \frac{1}{Ω_{B}^{2} - Ω^{2} - j Γ_{B} Ω} P_{12} (Ω) .

(9)

Here, Γ_B ∼ 2π × 30 MHz is the angular frequency linewidth of backward stimulated Brillouin scattering in standard fibers, ρ₀ is the density of silica , which is assumed to be the same for core and cladding, and the spatial overlap integral

Q_{E S}^{(c o r e)}

(m⁻¹) is defined as follows:

Q_{E S}^{(c o r e)} = 2 π \int_{0}^{\infty} [{\vec{e}}^{(c o r e) †} (r) {\vec{e}}^{(c o r e)} (r)] {\vec{e}}_{z}^{(a c)} (r) r d r .

(10)

In standard single-mode fibers, for P₁₂ = 1 W,

b^{(a c)} (Ω_{B})

equals about 3 × 10⁻²⁰ m². The stimulation of the dilatational, axial acoustic core mode is the most efficient at the Brillouin shift Ω_B. Stimulation of the acoustic core mode is accompanied by the coupling of optical power from the higher-frequency first pump wave to the lower-frequency second pump wave. Yet, in the following, we assume that the interaction is restricted to a comparatively short fiber section and that such an exchange of power remains small. We regard both the beating power

P_{12} (Ω)

and the acoustic modal magnitude

b^{(a c)} (Ω)

as independent of position z. This limitation will be lifted later in the discussion of spatially distributed sensing, in which at least one of the pump waves is modulated.

B. The scattering of an optical probe wave by Brillouin dynamic gratings

The material displacement of the acoustic core mode is associated with axial strain:

S_{z z} = \partial U_{z}^{(a c)} / \partial z

⁠. Strain, in turn, induces perturbations to the dielectric constant of the fiber medium, which propagate in the

\hat{z}

direction with angular frequency Ω and axial wavenumber q. The dielectric perturbations are proportional to the two-dimensional identity tensor I, and therefore they may be described by a scalar term,

\begin{align} Δ ε^{(a c)} (r, z, t) & = j q a_{2} b^{(a c)} (Ω) {\vec{e}}_{z}^{(a c)} (r) \exp (j q z - j Ω t) + c . c . \\ = - \frac{q^{2} a_{2}^{2}}{4 n_{eff}^{(c o r e)} c ρ_{0}} Q_{E S}^{(c o r e)} \frac{1}{Ω_{B}^{2} - Ω^{2} - j Γ_{B} Ω} \\ \times P_{12} (Ω) {\vec{e}}_{z}^{(a c)} (r) \exp (j q z - j Ω t) + c . c . \end{align}

(11)

The backward stimulated Brillouin scattering process is, therefore, associated with traveling periodic dielectric perturbations, namely, a moving grating, which propagates at the acoustic velocity v_L. Much like a stationary grating, the moving perturbations may scatter light. Due to the velocity of the moving gratings, we may expect a shift in the frequencies of scattered wave components. Unlike stationary gratings, the moving perturbations are dynamic; they may be switched on and off at will. Moreover, as will be discussed later, the judicious modulation of the two pump waves may confine the stimulation of these gratings to short fiber sections at arbitrary locations. Hence, the dielectric perturbations associated with the backward stimulated Brillouin scattering process came to be known as Brillouin dynamic gratings.^41,42

Figure 4(c) illustrates the dispersion relations in a scattering process between the core and cladding modes through a Brillouin dynamic grating. The illustration shows the single optical core mode in both directions, a cladding mode in the negative

- \hat{z}

direction, and the Brillouin dynamic grating, which follows the dispersion relation of the acoustic core mode. We consider an input probe field

{\vec{E}}^{(s)} (r, z, t)

in the optical core mode with an angular frequency ω_s that is higher than that of the first pump wave, ω_p1,

{\vec{E}}^{(s)} (r, z, t) = A_{s} {\vec{e}}^{(c o r e)} (r) \exp (j k_{s} z - j ω_{s} t) + c . c .

(12)

Here,

k_{s} = n_{eff}^{(c o r e)} ω_{s} / c

is the axial wavenumber of the probe wave, and A_s is the field magnitude (V). While coupling to cladding modes modifies this magnitude, we may safely assume that such coupling remains weak. We, therefore, consider A_s to be independent of z, similar to the pump waves.

The probe wave and the dielectric perturbation represented by the Brillouin dynamic grating induce nonlinear polarization, which includes a component of angular optical frequency ω_s − Ω,

\begin{align} {\vec{P}}^{(N L)} (r, z, t) & = ε_{0} Δ {ε^{(a c)}}^{*} {\vec{E}}^{(s)} \\ = - j ε_{0} q a_{2} {b^{(a c)}}^{*} (Ω) {\vec{e}}_{z}^{(a c) *} (r) A_{s} {\vec{e}}^{(c o r e)} (r) \\ \times \exp [j (k_{s} - q) z - j (ω_{s} - Ω) t] + c . c . \end{align}

(13)

The nonlinear polarization component propagates as a wave in the negative

- \hat{z}

direction, with a wavenumber k_s − q < 0. The nonlinear polarization may lead to the generation of light propagating in that direction with an angular frequency ω_s − Ω and the same wavenumber. This condition may be met for the m order cladding mode, provided that

q - k_{s} = \frac{n_{eff}^{(c o r e)} (2 ω_{p 1} - Ω)}{c} - \frac{n_{eff}^{(c o r e)} ω_{s}}{c} = \frac{n_{eff}^{(m)} (ω_{s} - Ω)}{c} .

(14)

The coupling of light to the cladding mode is wavenumber matched for a specific choice of the optical angular frequency of the probe wave in the core mode,

ω_{s, o p t}^{(m)} = \frac{2 n_{eff}^{(c o r e)} ω_{p 1} - (n_{eff}^{(c o r e)} - n_{eff}^{(m)}) Ω}{n_{eff}^{(c o r e)} + n_{eff}^{(m)}} \approx \frac{2 n_{eff}^{(c o r e)}}{n_{eff}^{(c o r e)} + n_{eff}^{(m)}} ω_{p 1} .

(15)

The detuning between the angular frequency of the pump wave and that of the probe wave at the optimal coupling to cladding mode m is given by

Δ ω_{s, o p t}^{(m)} = ω_{s, o p t}^{(m)} - ω_{p 1} \approx \frac{n_{eff}^{(c o r e)} - n_{eff}^{(m)}}{n_{eff}^{(c o r e)} + n_{eff}^{(m)}} ω_{p 1} .

(16)

For the cladding modes orders of the largest spatial overlap with the optical core mode, the angular frequency detuning corresponds to a wavelength difference of a few nanometers.^39,40 A more precise analysis of the wavenumber matching condition would also consider the variations of both effective indices with angular frequency. The reader is referred to more complete treatments.^58,59 The two effective indices in the denominator of Eq. would then be replaced by the respective group indices. The resulting correction in

Δ ω_{s, o p t}^{(m)}

is on the order of 3%. Similar to the Bragg condition for static gratings in Eq. , the spectrum of dynamic grating coupling to the cladding mode is offset by variations in

n_{eff}^{(m)}

due to the index of outside media, supporting chemical sensing opportunities.

Next, we may also quantify the relative fraction of the probe's optical power that is coupled to the cladding mode. To that end, the nonlinear polarization term of Eq. is used as the driving force of a nonlinear wave equation for a field component in the cladding mode,

{\vec{E}}^{(m)} (r, z, t) = A_{m} (z) {\vec{e}}^{(m)} (r) \exp [- j k_{m} z - j (ω_{s} - Ω) t] + c . c .

(17)

The magnitude of the field component is denoted by

A_{m} (z)

[V], and its wavenumber equals

k_{m} = n_{eff}^{(m)} (ω_{s} - Ω) / c

⁠. The negative sign of the wavenumber signifies propagation in the

- \hat{z}

direction. Note that the probe angular frequency ω_s does not necessarily match the optimum

ω_{s, o p t}^{(m)}

⁠. The nonlinear wave equation is of the form⁶⁰^,

\begin{align} \nabla^{2} {\vec{E}}^{(m)} (r, z, t) + \frac{{(ω_{s} - Ω)}^{2} n^{2} (r)}{c^{2}} {\vec{E}}^{(m)} (r, z, t) \\ = - \frac{{(ω_{s} - Ω)}^{2}}{ε_{0} c^{2}} {\vec{P}}^{(N L)} (r, z, t) . \end{align}

(18)

Since the cladding mode

{\vec{e}}^{(m)} (r) \exp [- j k_{m} z - j (ω_{s} - Ω) t]

is a solution to the linear, homogeneous wave equation, all terms on the left side of Eq. that do not involve z derivatives of the magnitude

A_{m} (z)

cancel out. We follow the slowly varying envelope approximation, stating that the axial evolution of the cladding mode magnitude is sufficiently gradual so that its second derivative term may be neglected. In addition, we multiply both sides of the nonlinear wave equation by

{\vec{e}}^{(m) †} (r)

and integrate over the transverse cross-section. The equation then reaches the form

\begin{align} \frac{d A_{m} (z)}{d z} & = - j \frac{ε_{0} a_{2}^{2} q^{2} ω_{s}}{2 n c ρ_{0}} Q_{E S}^{(c o r e) *} Q_{E S}^{(m) *} A_{p 1}^{*} A_{p 2} A_{s} \\ \times \frac{1}{Ω_{B}^{2} - Ω^{2} + j Γ_{B} Ω} \exp (j Δ k^{(m)} z) . \end{align}

(19)

In Eq. , we have approximated

n_{eff}^{(c o r e)} \approx n_{eff}^{(m)} \approx n

and

(ω_{s} - Ω) \approx ω_{s}

⁠. We have also defined the overlap integral between the transverse profiles of the optical core mode, the acoustic core mode, and the optical cladding mode (m⁻¹),

Q_{E S}^{(m)} = 2 π \int_{0}^{\infty} [{\vec{e}}^{(c o r e) †} (r) {\vec{e}}^{(m)} (r)] {\vec{e}}_{z}^{(a c)} (r) r d r .

(20)

The wavenumber mismatch term

Δ k^{(m)}

for the scattering of the probe wave to the cladding mode is given by

Δ k^{(m)} = q - k_{s} - k_{m} = \frac{n_{eff}^{(c o r e)} + n_{eff}^{(m)}}{c} (ω_{s} - ω_{s, o p t}^{(m)}) .

(21)

The mismatch vanishes at the probe's angular frequency of optimum coupling. A second nonlinear wave equation may be formulated for the evolution of A_s. The pair of equations would be coupled and could be solved jointly. However, as mentioned earlier, we assume that the coupling of power to the cladding mode is very weak, so that A_s is approximately fixed. At that limit, Eq. can be integrated directly. The optical power

P_{m} (z) = 2 n ε_{0} c {|A_{m} (z)|}^{2}

[W] that is coupled to the cladding mode, at a distance z from the launch end of the probe wave, equals

\begin{align} P_{m} (z, Ω, ω_{s}) & = {(\frac{a_{2}^{2} q^{2} ω_{s}}{4 n^{2} c^{2} ρ_{0} Ω_{B} Γ_{B}})}^{2} {|Q_{E S}^{(c o r e)}|}^{2} {|Q_{E S}^{(m)}|}^{2} \\ \times \frac{1}{1 + 4 \frac{{(Ω - Ω_{B})}^{2}}{Γ_{B}^{2}}} P_{1} P_{2} P_{s} \sin c^{2} (\frac{Δ k^{(m)} z}{2}) z^{2} . \end{align}

(22)

Here, P_1,2,s are the optical powers of the first pump wave, second pump wave, and optical probe wave, all taken to be fixed:

P_{1, 2, s} = 2 n ε_{0} c {|A_{1, 2, s}|}^{2}

⁠. In the expressions for optical power, we have again approximated

n_{eff}^{(c o r e)} \approx n_{eff}^{(m)} \approx n

⁠. The reflectivity of the Brillouin dynamic grating,

R_{m} (z) = P_{m} (z, Ω, ω_{s}) / P_{s}

⁠, depends on two angular frequency parameters: the offset Ω between the optical pump waves and the choice of the probe angular frequency. The dependence of the two is separable: the first represents electrostrictive stimulation and follows the Lorentzian line-shape that is characteristic of stimulated Brillouin scattering,^36,37 and the second follows the sinc shape of spontaneous scattering subject to wavenumber mismatch.⁶⁰ The solution for scattering to the cladding mode represents a four-wave mixing interaction driven by the third-order nonlinear phenomenon of opto-mechanics. In this case, the interaction takes place across two spatial optical modes.

Let us define a spatial efficiency factor

{|η^{(m)}|}^{2} = \frac{{|Q_{E S}^{(m)}|}^{2}}{{|Q_{E S}^{(c o r e)}|}^{2}},

(23)

and recall that the gain coefficient of backward stimulated Brillouin scattering within the single optical core mode, g_B (W⁻¹ × m⁻¹), is given by

g_{B} = \frac{a_{2}^{2} q^{2} ω_{s}}{4 n^{2} c^{2} ρ_{0} Ω_{B} Γ_{B}} {|Q_{E S}^{(c o r e)}|}^{2} .

(24)

In standard fibers, g_B ∼ 0.2 (W⁻¹ × m⁻¹). With those definitions, we may express the maximum power reflectivity in the cladding modes (Ω = Ω_B,

Δ k^{(m)} = 0

⁠), as follows:

\max \{R_{m} (z)\} = {|η^{(m)}|}^{2} g_{B}^{2} P_{1} P_{2} z^{2} .

(25)

The factor

{|η^{(m)}|}^{2}

⁠, therefore, represents the relative inefficiency penalty due to the transverse profile mismatch between core and cladding modes compared with standard, core mode-only Brillouin dynamic gratings. Figure 5 shows the calculated

{|η^{(m)}|}^{2}

as a function of modal order. The maximum relative efficiency is only 2.7%, obtained for m = 15. The spatial overlap integrals

Q_{E S}^{(m)}

vanish for even orders of the optical cladding modes up to m = 20. These modes carry very little light in the fiber core.²⁵ Although compromised in efficiency, Brillouin dynamic grating coupling to cladding modes is possible and useful, as demonstrated in Sec. IV. For a meter-long Brillouin dynamic grating and Watt-level pump waves,

\max \{R_{m} (z)\}

for the most efficient cladding modes is about 0.1%. For comparison, coupling to the same cladding modes through a strong permanent fiber Bragg grating may reach a power reflectivity of several percent.

FIG. 5.

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Calculated relative efficiency of Brillouin dynamic grating coupling of light from the optical core mode to optical cladding modes as a function of the cladding mode order.

The direct measurement of power that is coupled to a cladding mode is challenging. These modes often dissipate away before they might reach a detector at the end of the fiber. Alternatively, the probe wave in the core mode may be detected instead. Any coupling to a cladding mode would manifest in a loss of optical power in such a reading.

C. Spatially distributed analysis

Thus far, we have assumed both pump waves to be continuous. Therefore, Brillouin dynamic gratings could be formed over the entire length L of a fiber under test, and measurements of the output probe power would represent the spectrum of accumulated coupling, end to end. Local information cannot be retrieved in this manner. Spatially distributed analysis requires that the formation of Brillouin dynamic gratings be confined to short fiber sections at known locations in either transient or stationary manners.

Distributed sensing protocols based on Brillouin dynamic gratings follow those of standard Brillouin analysis in the core mode, which has been known for thirty years.^38,61 In Brillouin optical time-domain analysis, the intensity of the second pump wave, which counter-propagates with respect to the continuous probe, is modulated by repeating short and isolated pulses [Fig. 6(a)]. The first pump wave remains continuous. The Brillouin dynamic grating, therefore, propagates with the second pump in the form of a pulsed perturbation. Consider a fiber section located at a distance z from the launch point of the pulsed pump wave. We denote the time in which the pulse is launched as t = 0. The Brillouin dynamic grating is formed at that point following a delay t = z/v_g, where v_g is the group velocity of light in the fiber. The probe wavefront at position z may be attenuated at that time due to coupling to a cladding mode. The modified probe front reaches its output end of the fiber, the same position where the pump pulse is launched, following a two-way propagation delay of 2z/v_g. The detected probe power at that instance is unambiguously related to the coupling that took place at position z.

FIG. 6.

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Calculated magnitude of the acoustic wave in backward stimulated Brillouin scattering processes as a function of time and position along a fiber under test. (a): Time-domain analysis, in which the amplitude of one pump is modulated by a single pulse of 10 ns duration and the other pump is continuous. The acoustic perturbation propagates along the fiber alongside the pulsed pump and reaches different positions at different times.⁶² (b): Correlation domain analysis, in which the magnitudes of both pumps are fixed and they are jointly modulated by a binary phase sequence with a symbol duration of 0.1 ns. The magnitude of the acoustic wave is spatially confined to a narrow correlation peak of 1 cm width, where it is stationary, and its complex magnitude is rapidly oscillating about zero expectation value everywhere else.⁶³

The spatial resolution of the analysis is given by Δz = Δt · v_g/2, where Δt is the duration of the pump pulse. That duration, in turn, must exceed the lifetime of backward Brillouin stimulation 1/Γ_B, on the order of 5–10 ns.^36,37 The spatial resolution of the basic Brillouin optical time-domain analysis configuration is about 1 m.³⁸ Several advanced protocols successfully pushed that resolution to the centimeter scale.^38,64,65 These protocols, which will not be discussed here, are equally applicable to spatially distributed sensing through cladding modes that are excited by Brillouin dynamic gratings.

The power levels of the two pump waves, and hence the measurement signal-to-noise ratio, are limited by excessive Brillouin amplification or depletion, modulation instability, and the onset of amplified spontaneous Brillouin scattering.^66,67 These detrimental effects accumulate over the length of the fiber. Therefore, a longer measurement range would impose more stringent restrictions on pump power levels, and the signal-to-noise ratio would deteriorate accordingly.⁶⁸ Compared with standard Brillouin sensing protocols, the Brillouin dynamic grating cladding mode sensors face more severe limitations: Competing nonlinear mechanisms occur within the core mode with higher efficiency due to complete spatial overlap among all waves involved, whereas the coupling signal of interest is subject to the transverse inefficiency penalty of ${|η^{(m)}|}^{2}$ ⁠. The maximum reflectivity of the cladding modes, subject to the limits imposed by the nonlinear propagation of the pump waves, is inversely proportional to the number of resolution points squared: $\max \{R_{m}\} \propto {(Δ z / L)}^{2}$ ⁠.⁴⁰

Another protocol for the localization of the Brillouin dynamic gratings is that of optical correlation domain analysis.^{38,62,63,69,70} In this scheme, the magnitude of the two pump waves is fixed; however, the instantaneous phases or frequencies of both are jointly modulated by a common envelope function

f (t)

⁠. Due to the modulation, the magnitude of the stimulated acoustic wave

b^{(a c)} (Ω)

changes in time. Its expectation value

\bar{b^{(a c)}}

is proportional to the autocorrelation of the envelope function,

C_{f} [θ (z)]

⁠, where

θ (z) = (2 z - L) / v_{g}

is a position-dependent time lag between the two pump waves at point z,^{38,62,63,69,70}

\bar{b^{(a c)} (Ω, z)} = j \frac{q a_{2}}{4 n_{eff}^{(c o r e)} c ρ_{0}} Q_{E S}^{(c o r e)} \frac{1}{Ω_{B}^{2} - Ω^{2} - j Γ_{B} Ω} P_{12} (Ω) C_{f} [θ (z)] .

(26)

The stimulation of the Brillouin dynamic grating is, therefore, restricted to a section centered at the middle of the fiber, z = L/2,^{38,62,63,69,70} where

θ (z = L / 2) = 0

⁠. Note that the expression is identical to that of Eq. except for the autocorrelation factor. The axial extent of the grating equals half the correlation length of the envelope modulation function,

f (t)

⁠.^{38,62,63,69,70} The position of the dynamic grating is referred to as the correlation peak. At the peak location, the magnitude of the grating is fixed and equals that induced by continuous pump waves [Eq. ]. Outside the peak, the expectation value of the acoustic wave magnitude is much reduced.^{38,62,63,69,70} The measurement of the output probe intensity is, therefore, unambiguously related to the coupling to the cladding mode taking place at the correlation peak position. The correlation peak may be scanned along the fiber through control of the timing of the two input pump waves.^{38,62,63,69,70} Using correlation domain analysis, coupling to the cladding modes may be induced in arbitrarily located, short fiber sections in a random-access manner. For a detailed discussion of Brillouin optical correlation domain analysis, see Refs. 38 and 69–63.

Compared with time domain analysis, the correlation domain protocols can confine the stimulation of the acoustic waves to shorter sections, down to 1.6 mm in standard fiber, providing superior resolution.⁷¹ On the downside, point-by-point scanning of the correlation peak positions considerably extends the experimental duration, whereas time-domain analysis covers the entire length of the fiber in every trace. More elaborate variants of correlation domain analysis reduced the number of necessary position scans.⁷² Finally, note that both time-domain and correlation-domain analysis protocols must be repeated for multiple choices of the probe angular frequency to obtain a two-dimensional map of the local coupling spectra to the cladding modes.

IV. EXPERIMENTAL RESULTS OF SPATIALLY DISTRIBUTED OPTICAL CLADDING MODE SENSING

In this section, experimental methods and results of Brillouin dynamic grating coupling to the cladding modes of standard fibers are briefly presented. The reader is referred to recent references for complete detail.^39,40 A schematic illustration of the setup is shown in Fig. 7. The two pump waves were drawn from a common laser diode with a 1563.3 nm wavelength.^39,40 The exact optical frequency of the pumps was fine-tuned using a suppressed-carrier single-sideband (SC-SSB) electro-optic modulator driven by a microwave generator. The small-scale frequency offsets adjusted the difference between the optical frequencies of the pump waves and the probe field. The pump light was split into two branches. A first pump wave was amplified by an erbium-doped fiber amplifier to 30 dBm power and launched into one end of the fiber under test. The frequency of the second pump wave in the other branch was downshifted using a second SC-SSB modulator driven by a sine wave from a second microwave generator. The angular frequency difference Ω between the pump waves was chosen to match the Brillouin shift Ω_B ≈ 2π × 10.890 GHz of the fiber under test. The second pump wave was amplified by another fiber amplifier to 27 dBm power and launched from the opposite end of the fiber under test.

FIG. 7.

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Experimental setup for Brillouin optical time-domain analysis of dynamic grating coupling to the optical cladding modes of standard, single-mode fibers. BPF: tunable optical bandpass filter; SSB: single-sideband electro-optic modulator; EDFA: erbium-doped fiber amplifier; MZM: Mach-Zehnder electro-optic intensity modulator; Circ.: fiber-optic circulator.

A continuous probe wave of 13 dBm power was provided by another laser diode. Its wavelength could be tuned in 8 pm steps (1 GHz frequency steps) over several nanometers. Combined with the small-scale frequency offsets of the pump source mentioned earlier, the difference between pump and probe frequencies could be adjusted over a broad range with high accuracy. The probe was launched into the fiber under test in the same direction as the first pump wave. The Brillouin dynamic grating stimulated by the two pumps could couple light from the probe wave into counter-propagating cladding modes. The probe wave was detected by a photo-receiver at the output end of the fiber. A tunable optical bandpass filter blocked off the pump waves from reaching the detector.

The coupling to the cladding modes of the fiber under test was first characterized in a non-distributed manner. To that end, the second pump wave was amplitude-modulated by a sine wave of 50 kHz frequency. The strength of the stimulated Brillouin dynamic gratings was therefore modulated at the same rate. The dynamic grating coupling imposed a weak modulation at a 50 kHz rate on the detected power of the output probe wave. The modulation of the received signal was monitored by a lock-in amplifier. Figure 8(a) shows the measured and calculated spectra for a 50 cm-long, standard single mode fiber that was stripped off its protective coating and kept in air.³⁹ The lock-in signal exhibits multiple sharp peaks at specific probe wavelengths, which represent the coupling of the probe wave to cladding modes of consecutive odd orders between 11 and 21. Note that coupling to the even order modes vanishes, in agreement with calculations (Fig. 5). The observed wavelengths of probe wave coupling to the cladding modes agree very well with predictions.

FIG. 8.

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(a) Red: measured, normalized lock-in modulation signal of the output probe wave as a function of the probe wavelength. The pump’s wavelength is noted with a black dashed arrow. The standard, single-mode fiber under test was 50 cm long, stripped of its protective coating, and kept in air. The frequency difference between the two pump waves was adjusted to match the Brillouin frequency shift of the fiber. Multiple peaks are observed, corresponding to the coupling of the probe wave to cladding modes of odd orders between 11 and 21. Mode orders are noted above the spectral peaks. Blue: theoretical reflectivity spectrum. The measured wavelengths of peak coupling agree very well with the calculations.³⁹ (b) Measured normalized lock-in signal as a function of the detuning $Ω / (2 π)$ between the two pump waves and the difference $Δ ω / (2 π)$ between the pump and probe frequencies in the vicinity of the coupling peak to cladding mode m = 17. The offset $Δ ω / (2 π)$ of maximum coupling is designated as the zero point for convenience.³⁹ (c) Same as panel (a), for a meter-long section of fiber coated with a thin layer of fluoroacrylate polymer.⁴⁰

Figure 8(b) shows a two-dimensional scan of the normalized lock-in signal in the same fiber as a function of both Ω and ω_s near the peak of coupling to cladding mode m = 17.³⁹ The width of the coupling spectrum with respect to changes in Ω about Ω_B is 2π × 37 MHz, matching the Brillouin linewidth. The full width at half-maximum with respect to ω_s about $ω_{s, o p t}^{(17)}$ is 2π × 260 MHz. The expected width according to Eq. (22) is 2π × 185 MHz. The experimental value is somewhat broadened by sub-micron variations in the local cladding diameter along the fiber section under test. Figure 8(c) presents measured and calculated coupling spectra in a standard single-mode fiber that was coated with a thin layer of fluoroacrylate polymer with a refractive index n_coat of 1.405 RIU, below that of silica.⁴⁰ The coupling to the cladding modes of the coated fiber is evident. Here too, measurements agree well with calculations.

Figure 9(a) shows the distributed mapping of coupling spectra to cladding mode m = 17 along 2 m of bare fiber. The mapping was obtained using Brillouin optical correlation domain analysis: both pump waves were jointly modulated by a periodic binary phase sequence with a symbol duration of 800 ps.³⁹ The symbol duration corresponds to a spatial resolution of 8 cm. Two fiber sections, each 8 cm-long, were immersed in liquids: one in water and the other in ethanol. The probe frequencies of maximum coupling to the cladding mode are offset in both locations by 1.9 and 3.2 GHz, respectively.³⁹ The offsets quantitatively agree with expectations. The spatially distributed analysis of liquids outside the fiber cladding was thereby demonstrated. The presence of liquids could not be detected using Brillouin analysis in the core mode only. The experimental uncertainty in repeating measurements of $ω_{s, o p t}^{(17)}$ ⁠, in a fixed fiber position, was ±2π × 75 MHz.³⁹ That uncertainty corresponds to errors between ±4 × 10⁻⁴ and ±4 × 10⁻³ RIU in the estimate of n_ext, depending on its value. Variations in the coupling spectra among fiber positions kept in air are likely due to differences in the local cladding diameter, on the order of a few hundreds of nm.

FIG. 9.

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(a) Measured, normalized lock-in modulation signal of the output probe wave as a function of the frequency detuning $Δ ω / (2 π)$ between pump and probe and the position z of a localized Brillouin dynamic grating. The measurements were based on correlation domain analysis. Data in each z are normalized to a maximum of unity. Coupling to cladding mode m = 17 is observed in all positions. The bare fiber under test was kept in the air, except for two 8 cm-wide sections that were immersed in water and ethanol (see legend). The optimal frequency offsets at the two locations are shifted with respect to the baseline by 1.9 and 3.2 GHz, respectively.³⁹ (b) Detected voltage of the output probe wave as a function of position and probe wavelength offset. The measurements were based on time-domain analysis. The baseline difference between the pump and probe wavelengths was 2.4 nm. Coupling to a cladding mode is observed in five sections of fiber coated with a fluoroacrylate layer, each 1 m-long, separated by connectors with standard acrylate coating, in which cladding modes are not guided. All fibers were kept in the air. Coupling manifests in local dips in the output power of the probe wave in the core mode.³⁹ (c) Same as panel (b), for two coated fiber sections kept in air.⁴⁰ (d) Same as panel (c), 24 h following the immersion of the left coated fiber section only in acetone. The fiber section on the right served as a reference. The probe wavelength of maximum coupling to the cladding mode in the immersed section shifted by 10 pm, corresponding to a local change in the index outside the cladding by 0.047 RIU.⁴⁰

Figure 9(b) presents a distributed measurement of coupling spectra to cladding mode m = 13 along five sections of standard fiber.⁴⁰ The sections were 1 m-long each, and they were coated with a thin fluoroacrylate layer. Mapping was obtained through Brillouin optical time domain analysis: the magnitude of the second pump wave was modulated with repeating pulses of 10 ns duration, and the probe wave output power was measured as a function of time. The coupling with the cladding mode is evident in all sections. Differences among the exact wavelengths of maximum coupling are due to small scale variations in the cladding diameter and/or coating index.

Figures 9(c) and 9(d) present similar time-domain analyses of coupling spectra in two coated fiber sections connected in series.⁴⁰ In panel (c), both sections were kept in air, whereas in panel (d), only the left section was immersed in acetone for 24 h, and the right section served as a reference (see Ref. 40). The probe wavelength of peak coupling in the immersed section shifted with respect to the reference by 10 pm.⁴⁰ That offset corresponds to a change in the refractive index immediately outside the cladding of 0.047 RIU.⁴⁰ Smaller offsets were already observed within minutes following immersion.⁴⁰ The presence of acetone could be detected and localized based on its effect on the optical properties of the coating layer. The liquid could not be detected through measurements in the core mode alone.

V. RADIALLY SYMMETRIC ACOUSTIC CLADDING MODES OF STANDARD FIBERS

In the second part of this tutorial, we address the prospects of sensing outside the fiber using cladding modes that are acoustic rather than optical. In the previous discussion, acoustic waves were stimulated only in the core in order to scatter light into optical cladding modes. These modes, in turn, could monitor the refractive index outside the silica cladding boundary. We now rely yet again on optical pump waves in the core mode to stimulate acoustic waves. This time, however, the acoustic waves would propagate in cladding modes rather than core modes. The acoustic cladding modes are sensitive to the elastic properties of the surrounding media under test, and their oscillations are monitored by light in the single optical core mode.

A. Optical force induced by co-propagating pump tones

Brillouin scattering in the forward direction in optical fibers was first proposed, formulated, and demonstrated by Shelby, Levenson, and Bayer in 1985.³⁰ The process can be driven by optical forces induced by a pair of co-propagating pump tones in the optical core mode, with angular frequencies ω_1,2 = ω_p ± Ω/2. The frequency detuning

Ω / (2 π)

is on the order of hundreds of MHz. We consider pump tones that are linearly co-polarized, and we denote their common state of polarization as

\hat{x}

so that

{\vec{e}}^{(c o r e)} (r) = e^{(c o r e)} (r) \hat{x}

⁠. The wavenumbers of the two pump waves are

k_{1, 2} = n_{eff}^{(c o r e)} ω_{1, 2} / c

⁠, and their complex magnitudes are A_1,2. We assume, for the time being, that the pump waves are continuous and their magnitudes remain fixed along

\hat{z}

⁠. The two fields may be written as

{\vec{E}}_{1, 2} (r, z, t) = A_{1, 2} e^{(c o r e)} (r) \exp (j k_{1, 2} z - j ω_{1, 2} t) \hat{x} + c . c .

(27)

The two pump fields induce an electrostrictive force per unit volume that co-propagates with the two waves in the positive

\hat{z}

direction, with angular frequency Ω and wavenumber

q = n_{eff}^{(c o r e)} Ω / c

[Fig. 10(a)],⁷³^,

\begin{align} \vec{F} (r, ϕ, z, t) & = \frac{1}{4 n_{eff}^{(c o r e)} c} P_{12} (Ω) e^{(c o r e)} (r) \frac{\partial e^{(c o r e)} (r)}{\partial r} \\ \times [- (a_{1} + 2 a_{2}) \hat{r} - a_{1} \cos (2 ϕ) \hat{r} - a_{1} \sin (2 ϕ) \hat{ϕ}] \\ \times \exp (j q z - j Ω t) + c . c . \end{align}

(28)

Here ϕ is the transverse azimuthal coordinate with respect to the

\hat{x}

axis,

\hat{r}

and

\hat{ϕ}

denote the unit vectors in the radial and azimuthal directions, respectively, and a_1,2 are photoelastic parameters of silica.³² Their values are 0.66 and −1.19, respectively.³²

FIG. 10.

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(a) Dispersion relations between axial wavenumber and angular frequency in the optical core mode and in one radially symmetric acoustic cladding mode. The acoustic cladding mode is characterized by a cut-off frequency Ω_0m. Two co-propagating optical pump waves induce an electrostrictive force per unit volume. The angular frequency of the force is the difference between the two optical angular frequencies, and its wavenumber also equals the difference between the pump wavenumbers. The angular frequency and wavenumber of the force term may match those of the acoustic cladding mode close to its cut-off, where its phase velocity can equal that of the optical core mode. The stimulation of the acoustic cladding mode is accompanied by the coupling of optical power from the higher-frequency pump tone to the lower-frequency one. (b) Photoelastic scattering of an optical probe wave in the core mode by an acoustic cladding mode. The probe wave co-propagates with the two pump waves. It is significantly weaker than the pumps and does not contribute to the stimulation of the acoustic wave. Photoelastic scattering generates higher and lower sidebands and results in phase modulation.

The force propagates along the fiber with the phase velocity of guided light in the optical core mode. Compared with the case of counter-propagating pump tones, the wavenumber q of the force term in Eq. (28) is smaller by six orders of magnitude. The direction of the force vector is also different. The two co-propagating tones induce a force that is almost entirely transverse, whereas the counter-propagating pumps induce an axial force term (see Sec. III).

The transverse profile of the force consists of three components within the squared brackets of Eq. (28). The first term is radially symmetric and points in the radial direction $\hat{r}$ only. The remaining two terms depend on the azimuthal coordinate and follow a two-fold azimuthal symmetry. The radial symmetry of the fiber structure is removed by the direction of polarization of the pump waves. All three terms may contribute to the simulation of guided acoustic cladding modes, which maintain the same symmetries. The first force term may drive the oscillations of radial acoustic modes, whereas the latter two can generate torsional-radial modes of two-fold azimuthal symmetry.^30,31 Of the two classes of modes, the optomechanical interactions involving the radial modes are stronger and more widely investigated. While the sensing of media outside the cladding boundary can be performed using torsional-radial acoustic cladding modes as well,⁴⁴ we only consider the radial modes below for simplicity. The reader is referred to recent analyses of the more general case.^74,75

We express the radially symmetric force term in the following manner:

\vec{F} (r, z, t) = \frac{1}{4 n_{eff}^{(c o r e)} c} P_{12} (Ω) f_{0} (r) \exp (j q z - j Ω t) \hat{r} + c . c .,

(29)

with a transverse dependence function (m⁻³)

f_{0} (r) = - (a_{1} + 2 a_{2}) e^{(c o r e)} (r) \frac{\partial e^{(c o r e)} (r)}{\partial r} .

(30)

B. Stimulation of transverse acoustic cladding modes near cut-off and the modal linewidths

We consider next the guided acoustic cladding modes that may be stimulated by the force per unit volume term of Eq. (29). These modes are denoted by R_0m, where m is an integer. Each mode is characterized by a cut-off angular frequency Ω_0m, below which it may no longer propagate in the axial $\hat{z}$ direction [Fig. 10(a)]. As the acoustic angular frequency Ω approaches the cut-off value from above, the axial wavenumber becomes vanishingly small. The phase velocity of the acoustic modes, therefore, becomes arbitrarily high. Since the axial wavenumber q of the driving force is small, as noted earlier, the guided acoustic waves may be effectively stimulated only close to their cut-off. Therefore, we examine their properties at that limit. Note, however, that the acoustic axial wavenumber is not entirely zero, and the wave does propagate in the $\hat{z}$ direction.

We assume initially that the fiber is stripped of its polymer coating and kept in the air. Unlike the earlier solutions for the optical cladding modes, in the following analysis, we disregard the small-scale differences in elastic properties between core and cladding. The simplified analysis provides excellent agreement with the experiment, as shown in Sec. IV. The boundary condition for a bare fiber in air is zero radial stress at the cladding's outer radius r = a_clad.^30,76,77 The boundary condition may take the following form:^30,76,77

\frac{v_{S}^{2}}{v_{L}^{2}} J_{2} (\frac{Ω a_{clad}}{v_{L}}) = (1 - \frac{v_{S}^{2}}{v_{L}^{2}}) J_{0} (\frac{Ω a_{clad}}{v_{L}}) .

(31)

Here, v_L,S denote the velocities of dilatational and shear acoustic waves in silica, respectively, and J_0,2 are the Bessel functions of the first kind, orders zero and two, respectively. The solutions to the equation are the modal cut-off angular frequencies, Ω_0m.

The material displacement vector of the R_0m modes includes both radial and axial components. However, close to the cut-off, the axial displacement component becomes negligible. We, therefore, consider the material displacement to be entirely radial, with a transverse profile

{\hat{e}}_{r}^{(a c, m)} (r)

(m⁻¹). The transverse profile is normalized so that

2 π \int_{0}^{\infty} {|{\hat{e}}_{r}^{(a c, m)} (r)|}^{2} r d r = 1 .

(32)

It is explicitly given by

{\hat{e}}_{r}^{(a c, m)} (r \leq a_{clad}) = \frac{J_{1} (\frac{Ω_{0 m} r}{v_{L}})}{\sqrt{2 π \int_{0}^{a_{clad}} J_{1}^{2} (\frac{Ω_{0 m} r}{v_{L}}) r d r}} .

(33)

Here, J₁ is the first-order Bessel function of the first kind. The displacement vector

{\vec{U}}^{(a c, m)}

(m) is expressed as

{\vec{U}}^{(a c, m)} (r, z, t) = b^{(a c, m)} (Ω) {\hat{e}}_{r}^{(a c, m)} (r) \exp (j q z - j Ω t) \hat{r},

(34)

where

b^{(a c \cdot m)} (Ω)

is a modal magnitude [m²]. Figure 11 shows the normalized transverse profile of mode R₀₆ with a cut-off angular frequency of 2π × 270 MHz.

FIG. 11.

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Normalized transverse profile of radial material displacement in radially symmetric acoustic cladding mode R₀₆ of a standard bare fiber with a cladding diameter of 125 µm. The cut-off angular frequency Ω₀₆ of the mode is 2π × 270 MHz.

The magnitude of the stimulated modal oscillation scales with the electrostrictive driving force,

b^{(a c, m)} (Ω) = \frac{1}{4 n_{eff}^{(c o r e)} c ρ_{0}} P_{12} (Ω) Q_{E S}^{(m)} \frac{1}{Ω_{0 m}^{2} - Ω^{2} - j Ω Γ_{0 m}} .

(35)

Here, we have defined the spatial overlap integral between the transverse profiles of the modal displacement and the force per unit volume (with units of m⁻²),

Q_{E S}^{(m)} = 2 π \int_{0}^{\infty} f_{0} (r) {\hat{e}}_{r}^{(a c, m)} (r) r d r .

(36)

The stimulation of each radial mode is strongest at the cut-off angular frequency Ω_0m.

As an alternative to stimulation by a pair of pump tones, the acoustic cladding modes may also be driven by pump pulses of instantaneous power $P (t)$ (W). Let us denote the Fourier transform of the instantaneous pump power as $\tilde{P} (Ω)$ ⁠, in units of W × rad⁻¹ × Hz⁻¹. The magnitude of modal oscillations may be expressed in the angular frequency domain, $\tilde{b}$ ^(ac,m)(Ω), using the form of Eq. (36) with $P_{12} (Ω)$ replaced by $\tilde{P} (Ω)$ ⁠.⁷³ The units of $\tilde{b}$ ^(ac,m)(Ω) are m² × rad⁻¹ × Hz⁻¹. A packet of multiple acoustic cladding modes may be stimulated by short, broadband pump pulses.⁷⁵

The modal linewidth Γ_0m quantifies the losses of the acoustic cladding modes, and it is central to the comparatively new concept of sensing based on these modes. One source of loss is internal dissipation within the core and cladding of the optical fiber. These losses are relatively small. They are quantified by internal linewidth contributions, $Γ_{0 m}^{(i n t)}$ ⁠, that are mode dependent. The internal contributions are the only linewidths terms when a bare standard fiber is kept in air. They increase with modal order and scale quadratically with Ω_0m.⁷⁸ The linewidth may also be broadened by inhomogeneities in the cladding diameter.⁷⁸ Typical values of $Γ_{0 m}^{(i n t)}$ are between 2π × 30 and 2π × 200 kHz.^43–54,78

The linewidths are significantly modified when the bare fiber is placed in a medium of finite mechanical impedance Z_ext (kg × m⁻² × s⁻¹). We consider the medium to extend outside the cladding indefinitely. This assumption is useful for most settings in immersion in liquids. The presence of an outside liquid medium alters the boundary conditions: the stress at the outer cladding radius must equal the liquid pressure.⁷⁹ In principle, a modified boundary condition equation can be formulated and solved to obtain a different discrete set of eigenvalues. These eigenvalues are complex: their real parts signify the cut-off frequencies of radial acoustic cladding modes of the fiber in liquid, and their imaginary parts represent modal linewidth contributions. While this solution is exact, it provides limited intuition. In addition, the differences in the cut-off angular frequencies between the solutions of the fiber in air or liquids are small. As an alternative to the exact solutions, we may assume that the cut-off frequencies remain the solutions of Eq. (31) and obtain excellent approximations for the modal linewidths through the following considerations.

Let us denote the coefficient of partial reflection of acoustic displacement at the boundary between the silica cladding and the liquid outside as ξ. The coefficient depends on the mechanical impedance of the liquid Z_ext,⁴³^,

ξ = \frac{Z_{SiO2} - Z_{ext}}{Z_{SiO2} + Z_{ext}},

(37)

where Z_SiO2 = 13.1 × 10⁶ (kg × m⁻² × s⁻¹) is the mechanical impedance of silica. Similar to the standard analysis of optical cavities, acoustic losses at the boundary may be represented in terms of an equivalent temporal decay time

τ^{(a c)}

⁠. The acoustic intensity is reduced by a factor of ξ² every round trip from the fiber axis to the outer edge of the cladding and back. The two-way propagation delay equals t_d = 2a_clad/v_L, leading to⁴³^,

ξ^{2} = \exp (- \frac{t_{d}}{τ^{(a c)}}) .

(38)

We may define the contribution of the outward dissipation to the modal linewidth as

Γ_{0 m}^{(e x t)} = 1 / τ^{(a c)}

⁠. Equations and lead to⁴³

Γ_{0 m}^{(e x t)} (Z_{ext}) = - \frac{v_{L}}{a_{clad}} \ln (\frac{Z_{SiO2} - Z_{ext}}{Z_{SiO2} + Z_{ext}}) .

(39)

The contribution

Γ_{0 m}^{(e x t)} (Z_{ext})

is the same for all modes, and it is typically much larger than

Γ_{0 m}^{(i n t)}

⁠. For liquids outside bare fiber, we may approximate the overall linewidth as

Γ_{0 m} (Z_{ext}) = Γ_{0 m}^{(e x t)} (Z_{ext}) + Γ_{0 m}^{(i n t)} \approx Γ_{0 m}^{(e x t)} (Z_{ext})

⁠. The dependence of the modal linewidths on the outside impedance underlies the principle of acoustic cladding mode sensors^43–54 (Fig. 12).

FIG. 12.

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Calculated decay rates of acoustic intensity $Γ_{0 m}^{(e x t)} / (2 π)$ in acoustic cladding modes as a function of the mechanical impedance of an infinite liquid outside the cladding boundary of a bare fiber.⁴⁸ The impedances of ethanol, water, and polyimide are noted by dashed vertical lines (left to right).

C. Photoelastic scattering by guided acoustic modes

The stimulated acoustic waves are seldom monitored directly in the elastic domain. Instead, they are observed through the counter-effect of the photoelastic scattering of light. The strain associated with the material displacement

{\vec{U}}^{(a c, m)}

induces photoelastic perturbations, which also propagate in the

\hat{z}

direction with angular frequency Ω and wavenumber q. The local perturbations may be expressed in terms of a 2 × 2 tensor

Δ ε^{(a c, m)}

of transverse components. In Cartesian transverse coordinates x, y, the tensor is given by^30,31,75

\begin{align} Δ ε^{(a c, m)} (r, ϕ, z, t) & = b^{(a c, m)} (Ω) (\begin{matrix} c o s^{2} (ϕ) μ_{0 m}^{r r} (r) + s i n^{2} (ϕ) μ_{0 m}^{ϕ ϕ} (r) & \cos (ϕ) \sin (ϕ) [μ_{0 m}^{r r} (r) - μ_{0 m}^{ϕ ϕ} (r)] \\ \cos (ϕ) \sin (ϕ) [μ_{0 m}^{r r} (r) - μ_{0 m}^{ϕ ϕ} (r)] & s i n^{2} (ϕ) μ_{0 m}^{r r} (r) + c o s^{2} (ϕ) μ_{0 m}^{ϕ ϕ} (r) \end{matrix}) \\ \times \exp (j q z - j Ω t) + c . c . = b^{(a c, m)} (Ω) μ^{(a c, m)} (r, ϕ) \exp (j q z - j Ω t) + c . c . \end{align}

(40)

Note that the components

Δ ε_{x x}^{(a c, m)}, Δ ε_{x y}^{(a c, m)}, Δ ε_{y x}^{(a c, m)}, Δ ε_{y y}^{(a c, m)}

are still expressed in polar coordinates r, ϕ. We denote the transverse dependence of the local perturbation tensor

Δ ε^{(a c, m)} (r, ϕ, z, t)

as

μ^{(a c, m)} (r, ϕ)

μ^{(a c, m)} (r, ϕ) = (\begin{matrix} c o s^{2} (ϕ) μ_{0 m}^{r r} (r) + s i n^{2} (ϕ) μ_{0 m}^{ϕ ϕ} (r) & \cos (ϕ) \sin (ϕ) [μ_{0 m}^{r r} (r) - μ_{0 m}^{ϕ ϕ} (r)] \\ \cos (ϕ) \sin (ϕ) [μ_{0 m}^{r r} (r) - μ_{0 m}^{ϕ ϕ} (r)] & s i n^{2} (ϕ) μ_{0 m}^{r r} (r) + c o s^{2} (ϕ) μ_{0 m}^{ϕ ϕ} (r) \end{matrix}),

(41)

with

μ_{0 m}^{r r} (r) = (a_{1} + a_{2}) \frac{\partial {\hat{e}}_{r}^{(a c, m)} (r)}{\partial r} + a_{2} \frac{{\hat{e}}_{r}^{(a c, m)} (r)}{r},

(42)

μ_{0 m}^{ϕ ϕ} (r) = a_{2} \frac{\partial {\hat{e}}_{r}^{(a c, m)} (r)}{\partial r} + (a_{1} + a_{2}) \frac{{\hat{e}}_{r}^{(a c, m)} (r)}{r} .

(43)

The effect of the photoelastic perturbations on guided light in the single optical core mode scales with the spatial overlap between the transverse profile

μ^{(a c, m)} (r, ϕ)

and that of the core mode,

{|e^{(c o r e)} (r)|}^{2}

⁠.^30,31,75 The position-averaged tensor is proportional to the identity tensor I,

\begin{align} \bar{Δ ε^{(a c, m)}} (z, t) & = b^{(a c, m)} (Ω) \exp (j q z - j Ω t) \\ \times \int_{0}^{\infty} μ^{(a c, m)} (r, ϕ) {|e^{(c o r e)} (r)|}^{2} r d r d ϕ + c . c . \\ = b^{(a c, m)} (Ω) \exp (j q z - j Ω t) Q_{P E}^{(m)} I + c . c . \end{align}

(44)

where we have defined the scalar spatial overlap integral

Q_{P E}^{(m)}

(m⁻²) between the transverse profiles of the dielectric perturbations and the optical core mode,^30,31,75

Q_{P E}^{(m)} = π (a_{1} + 2 a_{2}) \int_{0}^{\infty} (\frac{\partial {\hat{e}}_{r}^{(a c, m)} (r)}{\partial r} + \frac{{\hat{e}}_{r}^{(a c, m)} (r)}{r}) {|e^{(c o r e)} (r)|}^{2} r d r .

(45)

The scalar dielectric perturbations, of magnitude

b^{(a c \cdot m)} (Ω) Q_{P E}^{(m)}

⁠, correspond to traveling variations in the effective index,

Δ n^{(a c, m)} (z, t) \approx \frac{b^{(a c, m)} (Ω) Q_{P E}^{(m)}}{2 n} \exp (j q z - j Ω t) + c . c .

(46)

The index variations associated with the stimulated acoustic cladding modes can lead to the spontaneous scattering and modulation of optical probe waves in the core mode, which are not directly involved in the acoustic waves’ generation. The acoustic cladding modes can also couple optical power between the pair of initial pump fields

{\vec{E}}_{1, 2}

⁠. The latter process is referred to as forward stimulated Brillouin scattering.^30–33

We consider the spontaneous scattering of probe waves first. The process is illustrated in Fig. 10(b). The probe field in the optical core mode is the same as in Eq. ,

{\vec{E}}^{(s)} (r, z, t) = A_{s} {\vec{e}}^{(c o r e)} (r) \exp (j k_{s} z - j ω_{s} t) + c . c .,

(47)

with axial wavenumber

k_{s} = n_{eff}^{(c o r e)} ω_{s} / c

and magnitude A_s (V). The probe is significantly weaker than the pump tones, and it does not contribute to the stimulation of the acoustic cladding mode. The probe field and the photoelastic perturbations to the dielectric tensor represent nonlinear polarization terms of angular frequencies ω_s ± Ω,

\begin{align} {\vec{P}}_{+}^{(N L)} (r, ϕ, z, t) & = ε_{0} A_{s} b^{(a c, m)} (Ω) μ^{(a c, m)} (r, ϕ) {\vec{e}}^{(c o r e)} (r) \\ \times \exp [j (k_{s} + q) z - j (ω_{s} + Ω) t] + c . c ., \end{align}

(48)

\begin{align} {\vec{P}}_{-}^{(N L)} (r, ϕ, z, t) & = ε_{0} A_{s} {b^{(a c, m)}}^{*} (Ω) {μ^{(a c, m)}}^{*} (r, ϕ) {\vec{e}}^{(c o r e)} (r) \\ \times \exp [j (k_{s} - q) z - j (ω_{s} - Ω) t] + c . c . \end{align}

(49)

We assume once again that the probe wave is continuous and that its photoelastic scattering is weak and does not change its magnitude. We also consider the coupling of power between the two pump tones

{\vec{E}}_{1, 2}

to be small so that the magnitude of the stimulated acoustic mode

b^{(a c, m)} (Ω)

is also fixed for all fiber positions.

The nonlinear polarization terms give rise to the generation of probe sidebands

{\vec{E}}_{\pm}^{(s)} (r, z, t)

of angular frequencies ω_s ± Ω,

{\vec{E}}_{\pm}^{(s)} (r, z, t) = A_{\pm} (z) {\vec{e}}^{(c o r e)} (r) \exp [j k_{\pm} z - j (ω_{s} \pm Ω) t] + c . c .,

(50)

The wavenumbers of the two sidebands equal

k_{\pm} = n_{eff}^{(c o r e)} (ω_{s} \pm Ω) / c = k_{s} \pm q

⁠. The wavenumbers equal those of the nonlinear polarization terms of the same frequencies. Therefore, the generation of the probe sidebands through photoelastic scattering by acoustic cladding modes is inherently wavenumber-matched. The evolution of the sideband magnitudes

A_{\pm} (z)

along the fiber is governed by nonlinear wave equations⁶⁰^,

\nabla^{2} {\vec{E}}_{\pm}^{(s)} + \frac{{(ω_{s} \pm Ω)}^{2} n^{2} (r)}{c^{2}} {\vec{E}}_{\pm}^{(s)} = - \frac{{(ω_{s} \pm Ω)}^{2}}{ε_{0} c^{2}} {\vec{P}}_{\pm}^{(N L)} .

(51)

We substitute Eqs. – into the nonlinear wave equations, apply the slowly varying envelope approximations (see also Sec. III), multiply both sides by

{\vec{e}}^{(c o r e)} {(r)}^{†}

⁠, and integrate over the transverse cross-section. The pair of equations is then brought to the following form:⁷⁵

\frac{d A_{+} (z)}{d z} = j \frac{k_{0}}{2 n} b^{(a c, m)} (Ω) Q_{0 m}^{(P E)} A_{s},

(52)

\frac{d A_{-} (z)}{d z} = j \frac{k_{0}}{2 n} {b^{(a c, m)}}^{*} (Ω) Q_{0 m}^{(P E) *} A_{s} .

(53)

Here, k₀ = ω_s/c is the vacuum wavenumber at the probe wave frequency. Next, we substitute the acoustic modal magnitude

b^{(a c, m)} (Ω)

from Eq. and define the nonlinear optomechanical coefficient of the interaction between the optical core mode and the acoustic cladding mode R_0m,^75,80

\begin{align} γ_{0 m} (Ω) & = \frac{k_{0} Q_{P E}^{(m)} Q_{E S}^{(m)}}{8 n^{2} c ρ_{0}} \frac{1}{Ω_{0 m}^{2} - Ω^{2} - j Ω Γ_{0 m}} \\ \approx j \frac{k_{0} Q_{P E}^{(m)} Q_{E S}^{(m)}}{8 n^{2} c ρ_{0} Ω_{0 m} Γ_{0 m}} \frac{1}{1 + j \frac{2 (Ω_{0 m} - Ω)}{Γ_{0 m}}} . \end{align}

(54)

The modal nonlinear coefficient has units of (W⁻¹ × m⁻¹). It takes up a maximum value that is purely imaginary and positive at the acoustic cut-off angular frequency Ω = Ω_0m,

γ_{0 m} (Ω_{0 m}) \approx j \frac{k_{0} Q_{P E}^{(m)} Q_{E S}^{(m)}}{8 n^{2} c ρ_{0} Ω_{0 m} Γ_{0 m}} .

(55)

The nonlinear coefficient

|γ_{0 m} (Ω_{0 m})|

is plotted as a function of cut-off frequency

Ω_{0 m} / (2 π)

in Fig. 13. In bare fibers, it takes up a maximum value of about 10 W⁻¹ × km⁻¹ for mode m = 8, with a cut-off frequency near 370 MHz. This value is an order of magnitude smaller than g_B of backward Brillouin scattering due to the transverse profile mismatch between the optical core mode and acoustic cladding modes.

FIG. 13.

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Calculated peak nonlinear optomechanical coefficients $|γ_{0 m} (Ω_{0 m})|$ of radial acoustic cladding modes in a standard uncoated fiber as a function of the modal cut-off frequency $Ω_{0 m} / (2 π)$ ⁠.

With this definition, we may express the nonlinear wave equations as

\frac{d A_{+} (z)}{d z} = j γ_{0 m} (Ω) P_{12} (Ω) A_{s},

(56)

\frac{d A_{-} (z)}{d z} = j γ_{0 m} {(Ω)}^{*} P_{12} {(Ω)}^{*} A_{s} .

(57)

The integration of the two equations suggests that the two sidebands are of equal magnitude and scale linearly with length. The phases of the two sidebands, however, are different. Together, the two represent the phase modulation of the optical probe wave at angular frequency Ω.⁸¹ The phase modulation of the probe wave as a function of time and position z along the fiber is given by⁷⁵^,

Δ φ^{(a c, m)} (z, t) = γ_{0 m} (Ω) P_{12} (Ω) z \cdot \exp (j q z - j Ω t) + c . c .

(58)

The phase modulation adds up to that induced through the Kerr effect.⁸² Both mechanisms represent third-order nonlinear interactions induced by the pair of intense pump waves. The experimental characterization of photoelastic modulation of probe waves should often take the Kerr effect contribution into consideration.⁷⁵

When the fiber under test is long enough and/or the beating power between the pump waves is strong enough, the photoelastic scattering of the probe wave may lead to the cascaded generation of additional sidebands at angular frequencies of ω_s ± 2 Ω, ω_s ± 3 Ω, and so on. The evolution of the magnitudes of all sidebands would require a larger set of coupled nonlinear wave equations.⁸³ In this case too, however, the solution would still represent the phase modulation of the input probe, only with greater strength. The magnitudes of sidebands are given by the Jacobi-Anger series^81,83

A_{l} (z) = {(- 1)}^{l} J_{l} [γ_{0 m} (Ω) P_{12} (Ω) z] A_{s} .

(59)

Here, l is any integer, J_l is the Bessel function of the first kind order l, and A_l is the magnitude of the modulation sideband at angular frequency ω_s + lΩ.

The photoelastic phase modulation of the optical probe may also be rationalized in terms of the traveling grating of the refractive index perturbation $Δ n^{(a c, m)} (z, t)$ [Eq. (46)]. The index grating propagates in the positive $\hat{z}$ direction with the same phase velocity as the optical core mode. Therefore, the probe wave that enters the fiber at z = 0 and a given instance t is subject to the same index perturbation $Δ n^{(a c, m)} (z = 0, t)$ through the entire fiber length. The index perturbation leads to the accumulation of an extra phase of $k_{0} Δ n^{(a c, m)} (z = 0, t) \cdot z$ ⁠, which varies in time at frequency Ω. The magnitude of the phase modulation equals $γ_{0 m} (Ω) P_{12} (Ω) z$ ⁠. The characterization of photoelastic phase modulation spectra is the most common protocol in optical fiber sensors based on acoustic cladding modes.

The stimulation of acoustic cladding modes may also be monitored through the coupling of power between the two pump fields, which are repeated here for convenience,

{\vec{E}}_{1, 2} (r, z, t) = A_{1, 2} (z) e^{(c o r e)} (r) \exp (j k_{1, 2} z - j ω_{1, 2} t) \hat{x} + c . c .

(60)

This time, however, we allow for changes in

A_{1, 2} (z)

and

b^{(a c, m)} (z, Ω)

along the fiber. The two fields and the dielectric tensor perturbation

Δ ε^{(a c, m)}

give rise to nonlinear polarization terms of angular frequencies ω_1,2,

\begin{align} {\vec{P}}_{1}^{(N L)} (r, ϕ, z, t) & = ε_{0} A_{2} (z) b^{(a c, m)} (z, Ω) μ^{(a c, m)} (r, ϕ) {\vec{e}}^{(c o r e)} (r) \\ \times \exp (j k_{1} z - j ω_{1} t) + c . c ., \end{align}

(61)

\begin{align} {\vec{P}}_{2}^{(N L)} (r, ϕ, z, t) & = ε_{0} A_{1} (z) {b^{(a c, m)}}^{*} (z, Ω) {μ^{(a c, m)}}^{*} (r, ϕ) {\vec{e}}^{(c o r e)} (r) \\ \times \exp (j k_{2} z - j ω_{2} t) + c . c . \end{align}

(62)

The wavenumbers of the nonlinear polarizations match those of the respective fields. The evolution of

A_{1, 2} (z)

is described through a pair of nonlinear wave equations for

{\vec{E}}_{1, 2}

⁠. Following the same procedure as mentioned earlier, the two equations are brought to the form⁷⁵

\frac{d A_{1} (z)}{d z} = j γ_{0 m} (Ω) P_{12} (Ω) A_{2} (z),

(63)

\frac{d A_{2} (z)}{d z} = j γ_{0 m} {(Ω)}^{*} P_{12} {(Ω)}^{*} A_{1} (z) .

(64)

The pair of equations can be modified to describe the evolution of the optical power levels of the two pump waves,

P_{1, 2} (z) = 2 n_{eff}^{(c o r e)} ε_{0} c {|A_{p 1, 2} (z)|}^{2}

(W),

\frac{d P_{1} (z)}{d z} = - α P_{1} (z) - 2 I m \{γ_{0 m} (Ω)\} P_{1} (z) P_{2} (z),

(65)

\frac{d P_{2} (z)}{d z} = - α P_{2} (z) + 2 I m \{γ_{0 m} (Ω)\} P_{1} (z) P_{2} (z) .

(66)

The above-mentioned equations were extended to include linear losses with a coefficient α (m⁻¹). Note that the imaginary part of

γ_{0 m} (Ω)

is positive for all acoustic angular frequencies [see Eq. ]. The pair of coupled nonlinear wave equations, therefore, describe the coupling of optical power between the two pump tones, from the higher-frequency one to the lower-frequency one, which accompanies the stimulation of the acoustic cladding mode [Fig. 10(a)]. The equations can be rearranged as

\begin{align} I m \{γ_{0 m} (Ω)\} & = \frac{1}{4 P_{1} (z) P_{2} (z)} \\ \times \{\frac{d}{d z} [P_{2} (z) - P_{1} (z)] + α [P_{2} (z) - P_{1} (z)]\} . \end{align}

(67)

This relation is used in one of the protocols for spatially distributed analysis of coupling spectra to acoustic cladding modes (see below).

The pair of Eqs. and may be solved analytically,⁸⁴

P_{1} (z) = P_{1} (0) \frac{1 + M^{- 1}}{1 + M^{- 1} \exp [g_{0 m} (Ω) z_{eff}]} \exp (- α z),

(68)

P_{2} (z) = P_{2} (0) \frac{1 + M}{1 + M \exp [- g_{0 m} (Ω) z_{eff}]} \exp (- α z) .

(69)

Here, we have used the following definitions:

M = P_{1} (0) / P_{2} (0)

is the ratio of input power levels between the two tones,

g_{0 m} (Ω) = 2 P_{1} (0) I m \{γ_{0 m} (Ω)\} (1 + M^{- 1})

(m⁻¹) is a gain coefficient, and

z_{eff} = [1 - \exp (- α z)] / α

is the effective length (m).

The solution is precise only for relatively weak coupling, $g_{0 m} (Ω) z_{eff} ≪ 1$ ⁠. For sufficiently large pump power levels and/or fiber lengths, the coupling of power between the two pump tones is accompanied by the generation of modulation sidebands of increasing order, as described earlier. In that case, the pair of nonlinear wave equations must be extended to a larger set, and the evolution of the two input tones deviates from that of Eqs. (68) and (69).⁸² Note that the nonlinear coefficient of the Kerr effect is purely real-valued; hence, it does not contribute to the coupling of power between the pump waves.⁸² By contrast, the generation of modulation sidebands takes place through photoelastic scattering and the Kerr effect combined, similar to the aforementioned phase modulation of a probe wave in the spontaneous regime.⁸² The monitoring of pump power levels can also provide the basis for spatially distributed analysis of acoustic cladding mode spectra.

D. Coated fibers and spatially distributed analysis

Thus far, it has been assumed that the fiber was stripped of any polymer protective coatings. However, the application of bare fibers outside the research laboratory is impractical. The presence of polymer coating layers may alter the preceding analysis considerably. Many types of coating, such as the highly popular dual-layer acrylates, absorb the stimulated acoustic waves that radiate out of the cladding before they may reach their outer boundary. In modeling acoustic cladding modes in such fibers, the coating may be considered infinite. The direct sensing of media beyond these coatings is not possible. The modal linewidth Γ_0m is broadened by the presence of the coating due to the dissipation of acoustic radiation according to the mechanical impedance of the coating layer. Sensing may be carried out indirectly in cases where a target reagent modifies the elastic properties of the coating layer itself,⁵⁰ similar to the optical cladding mode sensors discussed in earlier sections.

Unlike the case of optical cladding modes, certain polymer coating layers are sufficiently thin and have sufficiently low absorption to allow for the propagation of acoustic waves to their outer boundaries. These include polyimide and fluoroacrylate coating layers. When using these layers, the mechanical impedance of a liquid outside the coating boundary modifies the acoustic cladding mode spectra and may be monitored directly. This opportunity is not provided with the optical cladding modes of coated fibers. However, these measurements do raise difficulties that are not encountered when using bare fibers.

Without the coating, we have established a one-to-one, universal relation between the mechanical impedance Z_ext of a liquid substance under test and the modal linewidth Γ_0m. That relation was independent of the modal order m. Moreover, the effect of the exact cladding radius a_clad on the modal linewidth was small. If an intermediate coating layer of finite thickness is placed between the cladding and the outside medium, the spectra become markedly different (see Fig. 14). The modal linewidths for a given medium vary strongly among modes and heavily depend on small changes in coating thickness, well within specified tolerances. The linewidths of acoustic cladding modes for a given fiber section under test and a given liquid outside the coating cannot be predicted a priori. The specific modes used in the analysis should be chosen carefully, and pre-calibration on known test media is mandatory. However, quantitative sensing of liquids using the acoustic cladding modes of commercially available coated fibers has been achieved, as shown in Sec. VI.^46,48

FIG. 14.

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Calculated linewidths due to acoustic dissipation out of standard single-mode fibers coated with a thin layer of polyimide as functions of the mechanical impedance of a liquid medium outside the coating.⁴⁸ (a) Linewidths $Γ_{05}^{(e x t)} / (2 π)$ of acoustic mode R₀₅ (cut-off frequency near 178 MHz). (b) Linewidths $Γ_{09}^{(e x t)} / (2 π)$ of acoustic mode R₀₉ (cut-off frequency near 325 MHz). In each panel, calculation results are shown for several outer radii of the coating layer (see legends). The expected linewidths for a bare fiber are plotted as well for comparison. The linewidths for a given medium outside the coating vary among modes and with the exact coating radius. The geometric sensitivity of mode R₀₅ is much lower than that of mode R₀₉. The mechanical impedances of ethanol, water, and polyimide are noted by vertical dashed lines in both panels. Licensed under CC BY.

Finally, changes to the coating layer itself may be of interest, for example, in the research and development of materials, in production line quality control, or in preventive maintenance.^49,50 The temperature dependence of acoustic velocities in coating layers has been mapped.⁴⁹ Gamma radiation was successfully detected and quantified based on its effect on the acoustic velocity in a thin layer of fluoroacrylate coating,⁵⁰ the same layer that also served in optical cladding modes monitoring immersion in acetone (Ref. 40, Sec. IV).

A major challenge for fiber sensors based on acoustic cladding modes has to do with their extension toward spatially distributed analysis. As shown earlier, the photoelastic scattering induced by these modes takes place in the forward direction. The scattering spectrum may be readily observed at the far end of the fiber. However, the location of a specific contribution of forward scattering may not be directly identified through time-of-flight analysis, which underlies the distributed mapping of backscatter.⁵ Most solutions proposed to address this challenge have been indirect: the forward scattering spectra have been deduced through the analysis of an auxiliary backscattering mechanism.

In one example, the optical time domain reflectometry of Rayleigh backscatter has been extended to dual-frequency operation.⁴⁷ The measurements estimated the local power levels of two pump pulses, $P_{1, 2} (z)$ ⁠, through the analysis of collected Rayleigh backscatter in both time and frequency.⁴⁷ The collected data were differentiated with respect to position to obtain an estimate of $d [P_{2} (z) - P_{1} (z)] / d z$ ⁠. With that estimate, the local value of $I m \{γ_{0 m} (Ω)\}$ could be evaluated using Eq. (67).⁴⁷ In a different protocol, the magnitudes of multiple sidebands of a phase modulated probe wave, $A_{l} (z)$ ⁠, were estimated through Brillouin optical time domain analysis.⁴⁵ Each sideband served as a pump wave for the backward stimulated Brillouin scattering process, and its local magnitude $A_{l} (z)$ could be estimated through measurements of an auxiliary backward Brillouin signal.⁴⁵ With the mapping of multiple sidebands, the extent of phase modulation accumulated up to position z could be evaluated using Eq. (59). Here too, the obtained trace was differentiated with respect to the position to retrieve the local $γ_{0 m} (Ω, z)$ ⁠.⁴⁵

Of the two indirect methods described earlier, the measurements assisted by backward Brillouin scattering have been more successful. The spatial resolution has been recently brought below 1 m, with over 200 resolution points.^53,54 Yet performance remains modest. The differentiation of collected traces with respect to position is highly prone to noise and remains a drawback of indirect mapping protocols. The direct distributed analysis of forward scattering spectra has recently been achieved for the first time using the acoustic cladding modes of polarization maintaining fibers.⁵¹ The mapping relied on inter-modal scattering between the two non-degenerate principal axes of the fiber.^51,85 There as well, however, the resolution was limited to 60 m with only 17 resolved locations.⁵¹ Further breakthroughs are necessary to obtain technologically viable distributed fiber sensing based on acoustic cladding modes.

Point-sensing of media outside a centimeter-long section of bare fiber, based on acoustic cladding modes, has been demonstrated last year.⁵² The measurement relied on the stimulation of acoustic cladding modes by pump waves in one core of a multi-core fiber and on monitoring the induced photoelastic perturbations to the reflectivity of a Bragg grating in a different core.⁵² In a series of recent studies by Sanchez et al., the local spectra of both radial and torsional-radial acoustic cladding modes were measured using long period gratings.^86–89 The technique served for precise estimates of the fiber’s Poisson ratio⁸⁷ and simultaneous and unambiguous measurements of temperature and strain.⁸⁸

VI. EXPERIMENTAL RESULTS OF ACOUSTIC CLADDING MODE SENSING

An experimental setup for acoustic cladding modes fiber sensing, based on a Sagnac interferometer loop, is illustrated in Fig. 15(a).⁹⁰ The technique was first proposed and employed by Kang and co-workers in 2009.⁹⁰ The section of fiber under test is placed within the interferometer loop. Light from a first laser diode in the 1550 nm wavelength range is intensity modulated in an electro-optic Mach-Zehnder modulator to obtain pump pulses of nanosecond-scale duration and a period of several microseconds. The pump pulses are amplified by an erbium-doped fiber amplifier to average power levels of hundreds of mW and launched into the section under test in the clockwise direction only. The pump pulses stimulate wave packets of guided acoustic modes (see Sec. V B and Ref. 75). A polarization scrambler is used to suppress the contributions of non-radial modes.⁷⁴ A tunable optical bandpass filter blocks the pump pulses from reaching the loop output.

FIG. 15.

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(a) Schematic illustration of an experimental setup for the measurement of scattering spectra by acoustic cladding modes based on a Sagnac interferometer loop.⁵⁰ EDFA: erbium-doped fiber amplifier; PC: polarization controller; EOM: electro-optic amplitude modulator. Licensed under CC BY. (b) Measured and calculated normalized optomechanical nonlinear coefficients $\sum_{m} |γ_{0 m} (Ω)|$ of scattering by radial acoustic cladding modes in a standard uncoated single-mode fiber.⁷⁴ The spectrum consists of a set of discrete and narrow peaks.

Continuous wave light of a different wavelength in the 1550 nm range from a second laser diode serves as an optical probe. The probe wave is coupled into the loop in both directions. The clockwise propagating probe wave replica acquires instantaneous phase modulation $Δ φ^{(a c, m)} (z = L, t)$ in the fiber section under test due to the stimulated acoustic waves and the Kerr effect combined. For most practical settings, $Δ φ^{(a c, m)} (z = L, t) ≪ 2 π$ ⁠. The counterclockwise propagating probe acquires much weaker photoelastic phase modulation due to the lack of wavenumber matching.

The beating of the two probe wave components at the loop output converts the non-reciprocal photoelastic phase modulation into an intensity signal. Polarization controllers within the loop and at the input path of the probe wave are adjusted to obtain maximum intensity variations at the loop output.⁹⁰ By contrast, environmental phase drifts along the fiber under test are common to both directions of propagation, and they are eliminated by the Sagnac loop arrangement.⁹⁰ The output probe wave is detected by a photo-receiver. The detected voltage is digitized by an oscilloscope at a rate of several giga-samples per second for further offline signal processing. Traces are typically averaged over thousands of repeating pump pulses to improve the measurement signal-to-noise ratio.

The phase modulation of the probe wave comprises of a Kerr effect contribution and photoelastic scattering combined. However, the contribution of the instantaneous Kerr effect vanishes as soon as the pump pulse ends, whereas phase modulation through forward Brillouin scattering continues over acoustic lifetimes of hundreds of nanoseconds or longer. Time gating of collected traces can remove the first few nanoseconds and eliminate the Kerr effect from the analysis of probe phase modulation. Figure 15(b) shows the measured and calculated spectra of scattering by stimulated radial acoustic cladding modes in a standard, uncoated single-mode fiber.⁷⁴ The spectrum consists of discrete peaks with linewidths on the order of 100–200 kHz. Measurements are in excellent agreement with expectations.

The spectra of stimulated acoustic modes were characterized using the same setup, with the bare fiber immersed in ethanol or water. The measured linewidths were used to estimate the mechanical impedance of the liquids using Eq. (39). Figure 16 (solid lines) shows the measured impedances as a function of the cut-off frequencies of the acoustic modes used in experiments.⁴³ The dashed lines denote the literature values of the liquids’ impedances.^91,92 The measurements retrieved the mechanical impedance with 1% accuracy. The results were the first demonstration of acoustic cladding mode sensing.⁴³ The technique can identify the presence of liquids outside certain types of protective polymer coatings. Figure 17 shows calculated and measured spectra with air, ethanol, and water outside a standard fiber coated by a thin layer of polyimide.⁴⁸ The liquids have a different effect on the linewidth of each mode. The modal response can be calibrated and accounted for. Similar results were obtained by Chow and Thevenaz.⁴⁶

FIG. 16.

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Solid lines—mechanical impedance of liquids estimated through the analysis of stimulated acoustic cladding modes as a function of the modal cut-off frequency.⁴³ Dashed lines—reference values.^91,92 Red (blue) traces correspond to a bare standard single-mode fiber in ethanol (water).

FIG. 17.

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Measured (solid) and calculated (dashed) scattering spectra of stimulated radial acoustic cladding modes in a standard single-mode fiber, coated by a thin layer of polyimide.⁴⁸ The fiber was kept in air (a), ethanol (b), and water (c).

Figure 18 shows an example of a spatially distributed analysis of acoustic cladding modes.⁴⁷ Measurements were obtained using pump pulse envelopes consisting of two tones detuned in angular frequency by a variable Ω. Rayleigh backscatter of the pump signal was collected and analyzed in both time and frequency and the local power levels of the two tones $P_{1, 2} (z)$ were estimated.⁴⁷ The local nonlinear optomechanical coefficient of the acoustic cladding mode was then evaluated using Eq. (67). The standard single-mode fiber under test was 3 km long, and it was coated through most of its length with a standard dual-layer acrylate coating. A 100 m long section, located 2 km from the input end, was coated instead with a thin layer of polyimide. The cut-off frequency $Ω_{07} / (2 π)$ of radial acoustic cladding mode R₀₇ is different in that section.

FIG. 18.

Measured nonlinear optomechanical coefficient of radial acoustic mode R07 as a function of frequency and position along 3 km of standard single-mode fiber under test.47 The fiber was coated with a standard dual-layer acrylate coating through most of its length. A 100 m-long section, located 2 km from the input end, was coated by a thin layer of polyimide instead. That section is characterized by a different modal cut-off frequency Ω07/2π. The spectra were mapped based on the estimated local exchange of optical power between a pair of pump tones within a pulsed envelope.47 The two local power levels were resolved using the analysis of Rayleigh backscatter in both time and frequency domains.47

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Measured nonlinear optomechanical coefficient of radial acoustic mode R₀₇ as a function of frequency and position along 3 km of standard single-mode fiber under test.⁴⁷ The fiber was coated with a standard dual-layer acrylate coating through most of its length. A 100 m-long section, located 2 km from the input end, was coated by a thin layer of polyimide instead. That section is characterized by a different modal cut-off frequency $Ω_{07} / (2 π)$ ⁠. The spectra were mapped based on the estimated local exchange of optical power between a pair of pump tones within a pulsed envelope.⁴⁷ The two local power levels were resolved using the analysis of Rayleigh backscatter in both time and frequency domains.⁴⁷

Figure 19 shows an example of spatially distributed sensing of liquids outside 730 m of standard fiber.⁴⁵ The fiber was coated through most of its length, except for a 30 m long section that was stripped of its polymer coating.⁴⁵ The measurements distinguish among air, ethanol, and water outside the bare fiber section. Here too, the acoustic cladding modes were stimulated by a pair of pump tones within a pulsed envelope.⁴⁵ A readout probe pulse was launched with a certain delay following the pump tones. The probe pulse was phase modulated by the stimulated acoustic cladding modes, as shown in Eq. (58). Due to the time lag between pump and probe pulses, the phase modulation of the probe wave was free of Kerr effect contributions.⁴⁵ The modulation induced upper and lower sidebands to the probe wave, separated by integer multiples of the acoustic angular frequency Ω [Eq. (59)]. The magnitudes of several sidebands were mapped as functions of fiber positions using standard backward Brillouin optical time-domain analysis.⁴⁵ Based on the sidebands’ magnitudes, the extent of phase modulation accumulated up to a point of interest z could be evaluated. The estimate was differentiated with respect to the position to obtain the local contribution of photoelastic scattering by acoustic cladding modes. The spatial resolution of sensing protocols assisted by backward Brillouin scattering analysis has since been enhanced to less than 1 m.^53,54

FIG. 19.

Measured relative nonlinear coefficients of forward Brillouin scattering through radial mode R07 as functions of acoustic frequency and position along a standard single-mode fiber under test.45 Spatially distributed mapping was carried out indirectly, using backward Brillouin scattering optical time-domain analysis. The 730 m long fiber was coated a with standard dual-layer acrylate coating, except for a 30 m long exposed section located 500 m from the input end. That section was kept in the air (a), immersed in ethanol (b), or immersed in water (c). The modal linewidths Γ07/2π of the local scattering spectra at that section were 1.81, 2.86, and 4.12 MHz in the three panels, respectively.45 Licensed under CC BY.

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Measured relative nonlinear coefficients of forward Brillouin scattering through radial mode R₀₇ as functions of acoustic frequency and position along a standard single-mode fiber under test.⁴⁵ Spatially distributed mapping was carried out indirectly, using backward Brillouin scattering optical time-domain analysis. The 730 m long fiber was coated a with standard dual-layer acrylate coating, except for a 30 m long exposed section located 500 m from the input end. That section was kept in the air (a), immersed in ethanol (b), or immersed in water (c). The modal linewidths $Γ_{07} / (2 π)$ of the local scattering spectra at that section were 1.81, 2.86, and 4.12 MHz in the three panels, respectively.⁴⁵ Licensed under CC BY.

VII. SUMMARY AND DISCUSSION

This tutorial has addressed a long-standing challenge for the optical fiber sensor community: how can one detect and analyze media outside the boundaries of the cladding and coating of standard fibers, where light in the single optical core mode cannot reach? One of the most successful solutions has been the use of cladding modes instead. The transverse profiles of cladding modes do reach the outer boundaries of the fiber cladding and perhaps even into coating layers. For the most part, sensing demonstrations have relied on the optical cladding modes of the fiber.^24,26–28 The properties of these modes may be affected by the refractive index of an outside substance. However, the coupling of light to and from the optical cladding modes has mandated the inscription of permanent fiber gratings, which restrict the sensors' operation to point measurements only. In recent years, the prospects of sensing using the acoustic cladding modes of the fibers have been introduced as well and have raised much interest.^43–54 The stimulation and monitoring of acoustic cladding modes may retrieve the elastic properties of the outside media under test.

The tutorial has been dedicated to two sensing schemes: spatially distributed analysis of coupling spectra to optical cladding modes and sensing based on acoustic cladding modes. Both share optomechanics as a common underlying theme. In the former case, light is coupled to an optical cladding mode by a moving grating of photoelastic perturbations, which is associated with a longitudinal acoustic wave in the core of the fiber. The acoustic wave is generated through a backward stimulated Brillouin scattering process between a pair of pump waves. Acoustic stimulation may be turned on and off at will and can be confined to short fiber sections at arbitrary locations. In the latter case, optical pump waves stimulate an acoustic cladding mode in a forward Brillouin scattering process. The acoustic waves, in turn, are monitored through the coupling of power between the pump waves and/or through the photoelastic scattering of an additional optical probe. Both techniques do not require specialty fibers, modifications to the structure of standard fibers, or the inscription of permanent gratings.

Sensing through both optical and acoustic cladding modes may be carried out with certain coated fibers but is subject to significant limitations. The guiding of optical cladding modes requires that the refractive index of the coating is lower than that of silica and that absorption in the coating is not exceedingly high. Many standard coating materials do not meet these requirements. In addition, the cladding modes of coated fibers thus far used in sensing demonstrations decay within the coating and do not reach its outer edge. Therefore, these modes do not probe the outer medium directly. Sensing using these optical cladding modes is indirect and may only rely on the effects of the surrounding medium on the index of the coating layer. However, optical cladding mode sensing using coated fibers has recently been demonstrated (see Sec. IV). Higher-order optical cladding modes do reach outside the coating boundary; however, their spatial overlap with the optical core mode is greatly reduced. So far, coupling to these modes has not been achieved.

The use of coated fibers in acoustic cladding modes sensors also raises challenges. Here too, certain types of coating absorb the acoustic waves before they reach the outer edge of the coating. Fibers coated with these layers would only support indirect acoustic cladding modes sensing through the effect of the surroundings on the elastic properties of the coating. Compared with the optical cladding modes, however, coating layers that transmit the acoustic cladding modes to their outer edges are more readily found. With these layers, the elastic properties of an outside substance can be probed directly in addition to the study of the coating layer itself.

The techniques presented here are complementary. They may be used to characterize both the optical and elastic properties of the medium under test. Together, they provide a novel and powerful fiber sensing toolset. In one recent example, we combined the two techniques to monitor the effect of acetone on a layer of fluoroacrylate coating outside a standard fiber.⁴⁰ We found that both the optical and elastic properties of the coating layer varied gradually over a time scale of hours. Immersion in acetone could not be identified by measurements in the optical core mode alone. Both sensing modalities described can become target-specific through proper functionalization of the cladding's outer interface or the coating layers. Most reported experiments were carried out with the fiber placed on a table or coiled up within a dish, and the contact with those surfaces did not compromise the sensing function. However, contact with a solid surface could potentially induce additional losses in optical cladding modes or modify the outward dissipation of acoustic cladding modes. The mounting of fibers in real-world sensing applications would require attention.

Of the two concepts, the use of the optical cladding modes is more sensitive and accurate, and it is more easily scaled to spatially continuous distributed analysis. The probe wave is coupled to an optical cladding mode in the opposite direction; hence, scattering events may be directly localized based on time-of-flight considerations. Protocols of distributed Brillouin optical analysis in the core mode are directly applicable to optical cladding mode sensing; both time-domain and correlation-domain analyses have been demonstrated (Refs. 39 and 40; see Sec. IV). The spatial resolution reached 8 cm. The measurements were sensitive enough to resolve changes to the index outside the cladding at the fourth decimal point.^39,40 Although only a short range of 5 m has been demonstrated thus far, the technique is simply scalable to hundreds of meters in length.

Sensing through acoustic cladding modes is more rudimentary. Many demonstrations thus far could only distinguish between air and liquid or between water and ethanol.^43,45–54 One experiment identified percent-level changes in the salinity of aqueous solutions,⁴³ and another successfully monitored the dissolved level of sucrose.⁴⁴ The main difficulty associated with acoustic cladding mode sensors is their distributed analysis: the forward scattering mechanism cannot be mapped directly based on time-of-flight. The most promising solution path to date has been the indirect mapping of local spectra using an auxiliary process of backward stimulated Brillouin scattering.^45,53,54 The spatial resolution has reached 80 cm, with over 200 resolution points.⁵⁴ Performance, however, remains modest. Local scattering contributions are identified through the differentiation of collected data with respect to position, a protocol that is highly susceptible to noise. The forward scattering spectra were successfully mapped in a direct manner using an inter-modal scattering process in polarization maintaining fibers.⁵¹ There too, only 17 fiber sections were resolved. Further innovation is required to turn the new modality of acoustic cladding modes fiber sensors into a technology.

In summary, recent advances in optical fiber sensors using optical and acoustic cladding modes have been introduced and explained. The concepts represent breakthroughs in addressing long-standing challenges in the optical fiber sensor community and hold promise for applications in chemical and biological sensing over long reaches. The reader is referred to a recent book⁷⁵ and to the other references given throughout the text for more detailed reports of specific aspects. The experimental setups and results, in particular, could only be discussed here in a concise form. We hope this tutorial will help introduce these new concepts of cladding mode sensing to students and professionals in fiber-optics and optomechanics and encourage them to invest their own research efforts in similar directions.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Avi Zadok: Conceptualization (lead); Formal analysis (supporting); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead). Elad Zehavi: Formal analysis (equal); Investigation (equal); Writing – review & editing (supporting). Alon Bernstein: Formal analysis (equal); Investigation (equal); Writing – review & editing (supporting).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

APPENDIX A: BOUNDARY CONDITIONS EQUATIONS AND TRANSVERSE PROFILES OF OPTICAL CLADDING MODES

The effective indices and the normalized transverse profiles of optical cladding modes are obtained through solutions to the boundary condition equation. The equation was derived and solved in an excellent reference by Erdogan.²⁵ The results are repeated here for completeness and the convenience of the readers.

To formulate the equations, let us first define the following parameters:

u_{1}^{2} = k_{0}^{2} (n_{core}^{2} - n_{eff}^{2}),

(A1)

u_{2}^{2} = k_{0}^{2} (n_{SiO2}^{2} - n_{eff}^{2}),

(A2)

w_{3}^{2} = k_{0}^{2} (n_{eff}^{2} - n_{ext}^{2}),

(A3)

u_{21} = \frac{1}{u_{2}^{2}} - \frac{1}{u_{1}^{2}},

(A4)

u_{32} = \frac{1}{u_{2}^{2}} + \frac{1}{w_{3}^{2}},

(A5)

J = \frac{J_{1}^{'} (u_{1} a_{core})}{u_{1} J_{1} (u_{1} a_{core})}

(A6)

K = \frac{K_{1}^{'} (w_{3} a_{clad})}{w_{3} K_{1} (w_{3} a_{clad})}

(A7)

p (r) = J_{1} (u_{2} r) N_{1} (u_{2} a_{core}) - J_{1} (u_{2} a_{core}) N_{1} (u_{2} r),

(A8)

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

(A9)

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

(A10)

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

(A11)

σ_{1} = j \frac{n_{eff}}{Z_{0}},

(A12)

σ_{2} = j n_{eff} Z_{0} .

(A13)

In the above-mentioned definitions, J₁ is the first-order Bessel function of the first kind, N₁ denotes the first-order Bessel function of the second kind, and K₁ stands for the first order modified Bessel function of the second kind. The prime notation represents the derivative of a function with respect to its argument, and Z₀ is the vacuum impedance.

Using these definitions, we may state the following two quantities:

ζ_{0} = \frac{1}{σ_{2}} \frac{u_{2} (J K + \frac{σ_{1} σ_{2} u_{21} u_{32}}{n_{SiO2}^{2} a_{core} a_{clad}}) p (a_{clad}) - K q (a_{clad}) + J v (a_{clad}) - \frac{1}{u_{2}} s (a_{clad})}{- u_{2} (\frac{u_{32}}{n_{SiO2}^{2} a_{clad}} J - \frac{u_{21}}{n_{core}^{2} a_{core}} K) p (a_{clad}) + \frac{u_{32}}{n_{core}^{2} a_{clad}} q (a_{clad}) + \frac{u_{21}}{n_{core}^{2} a_{core}} v (a_{clad})},

(A14)

ζ_{0}^{'} = σ_{1} \frac{u_{2} (\frac{u_{32}}{a_{clad}} J - \frac{n_{ext}^{2} u_{21}}{n_{SiO2}^{2} a_{core}} K) p (a_{clad}) - \frac{u_{32}}{a_{clad}} q (a_{clad}) - \frac{u_{21}}{a_{core}} v (a_{clad})}{u_{2} (\frac{n_{ext}^{2}}{n_{SiO2}^{2}} J K + \frac{σ_{1} σ_{2} u_{21} u_{32}}{n_{core}^{2} a_{core} a_{clad}}) p (a_{clad}) - \frac{n_{ext}^{2}}{n_{core}^{2}} K q (a_{clad}) + J v (a_{clad}) - \frac{n_{SiO2}^{2}}{n_{core}^{2} u_{2}} s (a_{clad})} .

(A15)

The effective indices n_eff of the discrete cladding modes are obtained through numerical solutions of the following equation:

ζ_{0} = ζ_{0}^{'} .

(A16)

With knowledge of the effective index, the radial profiles of the electric field

{\vec{e}}^{(m)}

(m⁻¹) may be expressed. To that end, we first define the following additional quantities, all evaluated at the effective index solution

n_{eff} = n_{eff}^{(m)}

⁠,

F_{2} = J - \frac{u_{21} σ_{2} ζ_{0}}{n_{core}^{2} a_{core}},

(A17)

G_{2} = ζ_{0} J + \frac{u_{21} σ_{1}}{a_{core}},

(A18)

F_{3} = - F_{2} p (a_{clad}) + \frac{1}{u_{2}} q (a_{clad}),

(A19)

G_{3} = - \frac{n_{ext}^{2}}{n_{SiO2}^{2}} [G_{2} p (a_{clad}) - \frac{n_{SiO2}^{2} ζ_{0}}{n_{core}^{2} u_{2}} q (a_{clad})] .

(A20)

Within the core, r ≤ a_core, the radial, azimuthal, and axial components:

e_{r}^{(m)} (r)

⁠,

e_{φ}^{(m)} (r)

⁠, and

e_{z}^{(m)} (r)

⁠, may be expressed as

\begin{align} e_{r}^{(m)} (r) = j C \frac{u_{1}}{2} \{J_{2} (u_{1} r) + J_{0} (u_{1} r) - \frac{σ_{2} ζ_{0}}{n_{core}^{2}} [J_{2} (u_{1} r) - J_{0} (u_{1} r)]\}, \end{align}

(A21)

\begin{align} e_{φ}^{(m)} (r) = \frac{u_{1}}{2} C \{J_{2} (u_{1} r) - J_{0} (u_{1} r) - \frac{σ_{2} ζ_{0}}{n_{core}^{2}} [J_{2} (u_{1} r) + J_{0} (u_{1} r)]\}, \end{align}

(A22)

e_{z}^{(m)} (r) = C \frac{u_{1}^{2} σ_{2} ζ_{0}}{n_{core}^{2} β} J_{1} (u_{1} r) .

(A23)

In the silica cladding, a_core ≤ r ≤ a_clad, the corresponding field components take up the forms

\begin{align} e_{r}^{(m)} (r) & = j C \frac{π u_{1}^{2} a_{core} J_{1} (u_{1} a_{core})}{2} \{- \frac{F_{2}}{r} p (r) + \frac{1}{u_{2} r} q (r) \\ - \frac{σ_{2}}{n_{SiO2}^{2}} [u_{2} G_{2} v (r) - \frac{n_{SiO2} ζ_{0}}{n_{core}^{2}} s (r)]\}, \end{align}

(A24)

\begin{align} e_{φ}^{(m)} (r) & = C \frac{π u_{1}^{2} a_{core} J_{1} (u_{1} a_{core})}{2} \{\frac{σ_{2}}{n_{SiO2}^{2}} [\frac{G_{2}}{r} p (r) - \frac{n_{SiO2}^{2} ζ_{0}}{n_{core}^{2} u_{2} r} q (r)] \\ + u_{2} F_{2} v (r) - s (r)\}, \end{align}

(A25)

\begin{align} e_{z}^{(m)} (r) = - C \frac{π u_{1}^{2} u_{2}^{2} σ_{2} a_{core} J_{1} (u_{1} a_{core})}{2 n_{SiO2}^{2} β} [G_{2} p (r) - \frac{n_{SiO2}^{2} ζ_{0}}{n_{core}^{2} u_{2}} q (r)] . \end{align}

(A26)

Finally, the field components in the outer medium, r ≥ a_clad, are given by

\begin{align} e_{r}^{(m)} (r) & = j C \frac{π u_{1}^{2} u_{2}^{2} a_{core} J_{1} (u_{1} a_{core})}{4 w_{3} K_{1} (w_{3} a_{clad})} \{- F_{3} [K_{2} (w_{3} r) - K_{0} (w_{3} r)] \\ + \frac{σ_{2} G_{3}}{n_{ext}^{2}} [K_{2} (w_{3} r) + K_{0} (w_{3} r)]\}, \end{align}

(A27)

\begin{align} e_{φ}^{(m)} (r) & = C \frac{π u_{1}^{2} u_{2}^{2} a_{core} J_{1} (u_{1} a_{core})}{4 w_{3} K_{1} (w_{3} a_{clad})} \{- F_{3} [K_{2} (w_{3} r) + K_{0} (w_{3} r)] \\ + \frac{σ_{2} G_{3}}{n_{ext}^{2}} [K_{2} (w_{3} r) - K_{0} (w_{3} r)]\}, \end{align}

(A28)

e_{z}^{(m)} (r) = C \frac{π u_{1}^{2} u_{2}^{2} a_{core} σ_{2} J_{1} (u_{1} a_{core})}{2 n_{ext}^{3} β K_{1} (w_{3} a_{clad})} G_{3} K_{1} (w_{3} r) .

(A29)

In the above-mentioned expression, β = k₀n_eff and J_0,2 are the Bessel functions of the first kind, orders zero and two, respectively. The modified Bessel functions of the second kind, orders zero and two, are noted by K_0,2. The magnitudes of all field components are multiplied by the common normalization unitless constant C, so that

2 π \int_{0}^{\infty} n (r) {\vec{e}}^{(m) †} (r) {\vec{e}}^{(m)} (r) r d r = 1 .

(A30)

APPENDIX B: BOUNDARY CONDITIONS EQUATIONS AND TRANSVERSE PROFILES OF ACOUSTIC CORE MODES

Acoustic core modes are guided by the contrast in elastic properties between the core and cladding of standard fibers. The modes of interest are largely confined to the core; hence, we regard the cladding as infinite. We denote the velocities of dilatational acoustic waves in the core and cladding as $v_{L}^{(c o r e)}$ and $v_{L}^{(c l a d)}$ ⁠, respectively. In germanium-doped cores of standard fibers, $v_{L}^{(c o r e)} < v_{L}^{(c l a d)}$ ⁠.^93,94 We assume that the differences in density and in velocity of acoustic shear waves between the two regions are negligible.⁹⁴ While these approximations take away from the generality of the modal solutions, they nevertheless provide useful estimates for the acoustic core modes of standard optical fibers.

The axial phase velocity

v_{L}^{(a c)}

of an acoustic mode that is predominantly dilatational in the axial direction is found through a solution to the following boundary condition equation:^93,94

\frac{h_{core} a_{core} J_{0}^{'} (h_{core} a_{core})}{J_{0} (h_{core} a_{core})} - \frac{h_{clad} a_{core} K_{0}^{'} (h_{clad} a_{core})}{K_{0} (h_{clad} a_{core})} = 0 .

(B1)

Here, J₀ is the Bessel function of the first kind, order zero, and K₀ denotes the modified Bessel function of the second kind, order zero. The prime superscript denotes the derivative of a function with respect to its argument. The parameters h_core, h_clad (m⁻¹), are defined as

h_{core} = Ω \sqrt{\frac{1}{{v_{L}^{(c o r e)}}^{2}} - \frac{1}{v_{L}^{(a c) 2}}},

(B2)

h_{clad} = Ω \sqrt{\frac{1}{v_{L}^{(a c) 2}} - \frac{1}{v_{L}^{(c l a d) 2}}} .

(B3)

Since

v_{L}^{(c o r e)} < v_{L}^{(a c)} < v_{L}^{(c l a d)}

⁠, both parameters are real-valued. With the phase velocity known, the radial profiles

{\vec{e}}_{z}^{(a c)} (r)

and

{\vec{e}}_{r}^{(a c)} (r)

of the axial and radial displacement components, respectively, may be expressed. In the core, r ≤ a_core, we find

{\vec{e}}_{z}^{(a c)} (r) = - j C J_{0} (h_{core} r),

(B4)

{\vec{e}}_{r}^{(a c)} (r) = C \frac{h_{core} v_{L}^{(a c)}}{Ω} J_{0}^{'} (h_{core} r) .

(B5)

In the cladding, r ≥ a_core,

{\vec{e}}_{z}^{(a c)} (r) = - j C \frac{J_{0} (h_{core} a_{core})}{K_{0} (h_{clad} a_{core})} K_{0} (h_{clad} r),

(B6)

{\vec{e}}_{r}^{(a c)} (r) = C \frac{J_{0} (h_{core} a_{core})}{K_{0} (h_{clad} a_{core})} \frac{h_{clad} v_{L}^{(a c)}}{Ω} K_{0}^{'} (h_{clad} r) .

(B7)

The transverse profile of the axial component is similar to that of the single optical core mode. In Eq. to , C is a normalization constant, chosen so that

2 π \int_{0}^{\infty} [{|{\vec{e}}_{r}^{(a c)} (r)|}^{2} + {|{\vec{e}}_{z}^{(a c)} (r)|}^{2}] r d r = 1 .

(B8)

In the guided acoustic core modes of interest to this work,

|{\vec{e}}_{r}^{(a c)} (r)| ≪ |{\vec{e}}_{z}^{(a c)} (r)|

⁠.

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Opto-mechanical fiber sensing with optical and acoustic cladding modes

I. INTRODUCTION

II. CLADDING MODES OF STANDARD OPTICAL FIBERS

III. COUPLING OF LIGHT FROM THE CORE MODE TO A CLADDING MODE USING BRILLOUIN DYNAMIC GRATINGS

A. Stimulation of Brillouin dynamic gratings by optical pump waves

B. The scattering of an optical probe wave by Brillouin dynamic gratings

C. Spatially distributed analysis

IV. EXPERIMENTAL RESULTS OF SPATIALLY DISTRIBUTED OPTICAL CLADDING MODE SENSING

V. RADIALLY SYMMETRIC ACOUSTIC CLADDING MODES OF STANDARD FIBERS

A. Optical force induced by co-propagating pump tones

B. Stimulation of transverse acoustic cladding modes near cut-off and the modal linewidths

C. Photoelastic scattering by guided acoustic modes

D. Coated fibers and spatially distributed analysis

VI. EXPERIMENTAL RESULTS OF ACOUSTIC CLADDING MODE SENSING

VII. SUMMARY AND DISCUSSION

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: BOUNDARY CONDITIONS EQUATIONS AND TRANSVERSE PROFILES OF OPTICAL CLADDING MODES

APPENDIX B: BOUNDARY CONDITIONS EQUATIONS AND TRANSVERSE PROFILES OF ACOUSTIC CORE MODES

REFERENCES

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Opto-mechanical fiber sensing with optical and acoustic cladding modes Open Access

I. INTRODUCTION

II. CLADDING MODES OF STANDARD OPTICAL FIBERS

III. COUPLING OF LIGHT FROM THE CORE MODE TO A CLADDING MODE USING BRILLOUIN DYNAMIC GRATINGS

A. Stimulation of Brillouin dynamic gratings by optical pump waves

B. The scattering of an optical probe wave by Brillouin dynamic gratings

C. Spatially distributed analysis

IV. EXPERIMENTAL RESULTS OF SPATIALLY DISTRIBUTED OPTICAL CLADDING MODE SENSING

V. RADIALLY SYMMETRIC ACOUSTIC CLADDING MODES OF STANDARD FIBERS

A. Optical force induced by co-propagating pump tones

B. Stimulation of transverse acoustic cladding modes near cut-off and the modal linewidths

C. Photoelastic scattering by guided acoustic modes

D. Coated fibers and spatially distributed analysis

VI. EXPERIMENTAL RESULTS OF ACOUSTIC CLADDING MODE SENSING

VII. SUMMARY AND DISCUSSION

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: BOUNDARY CONDITIONS EQUATIONS AND TRANSVERSE PROFILES OF OPTICAL CLADDING MODES

APPENDIX B: BOUNDARY CONDITIONS EQUATIONS AND TRANSVERSE PROFILES OF ACOUSTIC CORE MODES

REFERENCES

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Opto-mechanical fiber sensing with optical and acoustic cladding modes