Optical phase response to temperature in a hollow-core photonic crystal fiber

Seth Meiselman; Geoffrey A. Cranch

doi:10.1364/OE.25.027581

1. Introduction

Measurement of phase shifts in optical fiber using interferometry has been applied to various sensing applications ranging from inertial navigation sensors (i.e. gyroscopes) [1] to underwater hydrophones [2]. In gyroscopes, phase shifts are generated by the Sagnac effect and in hydrophones by displacements or strain applied to the fiber due to pressure. The fiber’s refractive index profile, length, and radius may also change as a result of a variation in temperature. These thermo-mechanical and thermo-optic effects cause a change in optical phase at the output of the fiber which is indistinguishable from the phase shift due to the Sagnac effect or mechanical perturbations.

In solid core silica optical fibers, the temperature response is determined by the thermo-optic coefficient of silica. One means of reducing the temperature dependence of the optical phase is to use hollow core optical fiber [3], where most of the light propagates in the air core and the thermal response is determined primarily by the thermal expansion properties of the surrounding silica material. Sensing based on hollow core optical fiber has demonstrated reduced thermal sensitivity in fiber optic gyroscopes [4], enhanced pressure response over conventional solid fibers [5] and different thermodynamic noise characteristics to solid core fibers [6].

Recently, a publication by Slavik et al., [7], reported thermal sensitivity in a hollow core fiber that was claimed to be substantially lower than that reported for a similar fiber in an earlier article, [8]. The disagreement in the literature regarding the thermal phase sensitivity of hollow core photonic bandgap fibers we believe stems from a mismatch in definitions used to compare the results of [7, 8] and that their results are broadly consistent with one another.

In section 2 we review the different definitions presented in [7, 8] and show how they should be related. To convince ourselves of the consistency of the previously reported findings, we measured the thermal phase response of four commercially available fibers: two hollow core photonic bandgap fibers, NKT HC19-1550 and NKT HC-1550-02; one hollow core (inhibited coupling) kagome lattice fiber, GLOphotonics PMC-C-TiSa_Er-7C; and standard solid core single mode fiber, Corning SMF-28. Section 3 presents a straight-forward way to measure the thermal phase response of the four fibers, the results obtained from the experiment, and shows that they are consistent with both [7] and [8].

In section 4 we use the method of [8] and extend it to include kagome-type lattice structured photonic crystal fibers and report numerically simulated results (shown with the data presented in section 3) for the three hollow core fibers mentioned above. We show that our numerical and experimental results are consistent with both [7] and [8].

Finally, in section 5 there is some further discussion of these results and the limitations of this method for the analytical modeling of thermal phase sensitivity for hollow core fibers.

2. Definition of sensitivity

There are applications where accurate timing and propagation delay are paramount for operation in a fiber network [9]. The time delay from propagating along a fiber of length L and group refractive index n_g is

τ = \frac{n_{g} L}{c},

where c is the speed of light in vacuum, [7,10,11]. In these applications, where thermal effects are of significant influence, an important measure or figure of merit for accurate transit times in a communication or sensor network would be the variation of optical delay with temperature per length of fiber,

\frac{1}{L} \frac{d τ}{d T} = \frac{1}{c L} (L \frac{d n_{g}}{d T} + n_{g} \frac{d L}{d T}) .

The first term is due to the thermo-optic effect and the second term is due to the effect of fiber elongation. As in [7], for fused silica with an operating wavelength of 1550 nm, we may approximate n_g with the phase index n.

Alternatively, there are applications that require precision of accumulated phase for accurate measurements. The accumulated optical phase of light with a propagation constant k, in a material of refractive index n, and traveling a length L is

ϕ = k n L .

Then a change in the accumulated phase as a function of a change in temperature follows as

\begin{array}{l} S_{ϕ} \equiv \frac{1}{ϕ} \frac{d ϕ}{d T} = \frac{1}{n} \frac{d n}{d T} + \frac{1}{L} \frac{d L}{d T}, \\ = S_{n} + S_{L} \end{array}

where this is a normalized thermal phase response or relative measure of the thermal phase sensitivity. The first term in the above expression is the thermally induced change in the refractive index due to the thermo-optic effect, the second term expresses the thermally induced change in length of the light guiding material. It is this term, S_ϕ, that is a figure of merit for a system that relies on phase measurements, [8, 12, 13]. In a more practical sense, (1/L) dϕ/dT is sometimes used instead.

To compare the change in delay and the change in phase with respect to temperature some conversion would be necessary. We suggest making an alteration to Eq. (2), and defining a normalized time delay response or relative thermal delay sensitivity as

\begin{array}{l} D_{τ} \equiv \frac{1}{τ} \frac{d τ}{d T} = \frac{1}{n} \frac{d n}{d T} + \frac{1}{L} \frac{d L}{d T}, \\ = D_{n} + D_{L} \end{array}

where it is now obvious that Eq. (4) and Eq. (5) are identical fiber responses. Then it is also clear that

\frac{1}{L} \frac{d τ}{d T} = (\frac{n}{c}) S_{ϕ} .

Now to fully address the stated discrepancy in the literature: [7] used (1/L) dτ/dT, whereas [8] used S_ϕ, the equivalent of D_τ, in their respective comparisons of data taken for hollow core photonic bandgap fibers in relation to conventional solid core single mode fiber. Whilst there is no standard definition for thermal sensitivity of an optical fiber, the appropriate definition will depend on the application. However, comparing the two figures of merit without appropriate scaling between them results in an over-estimation (∝ n_SMF/n_HC) by [7] in the ratio of thermal phase change of hollow core fiber to that of solid core single mode fiber compared to the ratio of thermal time delay change in two similar fibers.

In Table 1, results of both [7, 8] are presented along with appropriate conversion/scaling for a proper comparison. While the HC-PBGF presented in [7] shows a definite lower value of thermal sensitivity, Table 1 shows that the results presented by Slavik et al. are slightly overstated. The ratios for the fibers presented in [8] in terms of propagation delay (including measurement error) are between 4.9 and 19.6 for the Crystal Fibre AIR-10-1550, and between 4.4 and 7.8 for the Blaze Photonics HC-1550-02. We compare that to the ratios for the Crystal Fibre as between 3.3 and 13.2, and for the Blaze Photonics as between 2.7 and 5.3, in terms of the phase change or normalized delay. This supports our view that the results reported in the two papers are broadly consistent with another as the ratio of thermal sensitivity presented in [7] of SMF-28 fiber to the hollow core photonic bandgap fiber would be a factor of 18.7 in terms of propagation delay, whereas in terms of the normalized delay or phase change the ratio reduces to 12.8. Thus, the improvement in the results presented in [7] over those presented in [8] is overstated by approximately a factor of 1.45 (the ratio of the index of refraction for SMF-28 compared to air). Each of the authors of [7, 8] should be proud of their achievement and the general agreement of their reported results. The remaining differences between the two sets of data can be attributed to experimentally tested fiber’s different geometries, coatings, and measurement error, etc.

Table 1. Thermal phase response and time delay change for fibers presented in [7, 8]. Numbers in parentheses are derived from Eq. 6, assuming n = 1 for hollow core and n = 1.45 for solid core fibers, yielding scaling coefficient values of ∼ 3.33 and ~ 4.83, respectively. Reported length is in meters. Units for S_ϕ are in ppm/K, whereas units for (1/L)dτ/dT are given in ps/km/K.

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3. Experiment

To compare results reported in [8] and [7] with an independent check, we experimentally measured the thermally induced phase response of SMF-28 fiber, and two hollow-core photonic bandgap fibers: NKT HC19-1550 and NKT HC-1550-02 (the latter of which should agree with the Blaze Photonics fiber presented in [8]). Additionally we measured one hollow core inhibited coupling fiber, GLOphotonics PMC-C-TiSa_Er-7C, which has a kagome lattice photonic crystal structure. To make these measurements we constructed a fiber Mach-Zender interferometer and exposed one arm containing the fiber of interest to rapid heating via a cold/hot water bath.

The experimental setup, shown in Figure 1, consists of a RIO diode laser, λ = 1550 nm, whose output polarization is adjusted to maximize the interference signal before entering the fiber interferometer. The fiber interferometer is mostly contained in a lead-lined acoustic dampening aluminum box. The arm (Coil A) that remains in the acoustically shielded box is constructed from SMF-28 fiber and has a cylindrical lead-zirconate-titanate (PZT) piezo-actuator with multiple tight wrappings of the fiber to allow for frequency modulation used in calibrating the interferometer output. The other arm (Coil B) has a length of the fiber of interest spliced to SMF-28 patch fiber, is wound in a 6–7 cm diameter coil, and submerged in a cold water bath exterior to the acoustic shielding box. Locations of fiber splicing are shown in the figure. The output of interferometer was received by a single channel of a TTI TIA527 balance detector. The analog signal was then digitized by a National Instruments USB-6259-BNC data acquisition instrument and recorded by LabView software for analysis. Meanwhile, a thermocouple, National Instruments USB-TC01, was used to record the temperature of the water bath, next to the fiber coil, without causing a contact disturbance. Effort was taken to minimize the exposure of the patch fiber to air currents. Careful effort was also made to prevent the tested fiber from moving aggressively when the boiling water was added to the bath inducing the abrupt temperature change. The phase signal and temperature of the water bath were recorded for 90 seconds before the addition of boiling water to abruptly change the temperature. The phase signal and temperature were continually recorded for another 15 minutes to allow for turbulent motion in the water to subside, and the water bath to gradually cool.

Fig. 1 Experimental diagram for measuring the thermal response of hollow-core fibers, and SMF-28.

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By measuring the temperature change and corresponding phase change, we are able to calculate the normalized thermal response by taking

S_{ϕ} \to \frac{1}{ϕ} (\frac{Δ ϕ}{Δ t}) {(\frac{Δ T}{Δ t})}^{- 1}

where Δt is a long increment of time, 5 and 15 minutes, to avoid turbulence in the water bath after mixing (measured in the phase or in recorded temperature) and allow for longer steady cooling of the bath. The lengths of heated fiber are shown in Table 2, and the expected index of refraction is n_SMF = 1.458 for SMF-28, and n_HC ≃ 1 for the hollow-core fibers. Example temperature and phase data is shown in Fig. 2 for all fibers tested.

Table 2. Experimental thermal phase response, and numerical modeling of the response due to fiber elongation for the three hollow core fibers. Numbers in parentheses are derived from Eq. 6, assuming n = 1 for hollow core and n = 1.458 for solid core fibers, yielding scaling coefficient values of ∼ 3.33 and ∼ 4.83, respectively. Length is in meters. Units for S_ϕ and S_L are in ppm/K, and units for (1/L) dτ/dT are in ps/km/K.

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Fig. 2 Examples of fiber thermal phase response (blue) and corresponding temperature (orange) measurements. Clockwise from top left: large core photonic bandgap fiber, HC19-1550; large core inhibited coupling kagome type fiber, PMC-C-TiSa_Er-7C; solid core, SMF-28; small core photonic bandgap fiber, HC-1550-02.

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The values for the thermal phase response of the hollow core photonic bandgap fibers, HC19-1550 and HC-1550-02, found in Table 2 are between the two reports of [7] and [8] (see Table 1). We believe this supports the finding stated earlier in section 2, that in fact the measurements are broadly consistent with one another, and now, with the results presented here. The difference between the modeled S_L and the experimental value of S_ϕ for the SMF-28 fiber in the last column of Table 2 is of course due to S_n. It is interesting to note that the modeled value of S_L (details of which are in the next section) for the SMF-28 fiber closely matches to the reported value of the measured S_ϕ for the HC-PBGF presented by [7], possibly indicating that their fiber has approached a limit to the lowering of the thermal phase response. The value of the response for the inhibited coupling fiber is higher than that of the two photonic bandgap fibers. We believe this is the first reported result for the thermal phase response of this specific type of fiber structure. The reason for this increase is presented in the following section.

4. Thermo-mechanical sensitivity

In a solid-core fiber the refractive index term of the phase/delay response to a temperature change is much larger than the effect due to thermal expansion or compression. But the recent innovation of fibers with no solid material core that allows light to propagate along its axis in an air-core, allows for the minimization of that refractive index contribution. In an ideal hollow-core fiber where there is no optical power propagating outside the air-core, we should expect the refractive index term to be negligible in S_ϕ, and it can be shown that

S_{ϕ} \to S_{L} = \frac{ϵ_{z}}{Δ T} .

To examine how the thermally induced length change occurs in a hollow-core optical fiber we must make use of the geometry of the fiber and the mechanical relationships between the internal stresses and strains to determine the longitudinal strain, ϵ_z.

To solve for the longitudinal strain we follow the analytical method outlined in [8]. The basic method also extends to acoustic and other internal or external pressure sensitivity [5, 14–17]. We start with an ideal fiber where we assume the core is hollow, filled with a gas (presumably air) at pressure P₁, and that the exterior of the fiber is exposed to a gas or liquid at a pressure P₂. The layer next to the hollow core is a thin silica sheath that ranges from a ≤ r ≤ b. Then the silica-air honeycomb is from b ≤ r ≤ c with an air-filling ratio η (defined later). The final layer in the fiber is a silica cladding from c ≤ r ≤ d. In a practical setting the fiber would also have a soft polymer coating and or (multiple) jacketing layer(s), however, it has been shown before that if this is loosely coupled, the soft external material layer(s) present an insignificant addition to the signal we wish to observe here, so we will neglect it here. This fiber structure can be seen in Fig. 3.

Fig. 3 Transverse cross-section of hollow-core photonic bandgap fiber. The hollow core extends from the origin to r = a, an inner silica core-sheath from a ≤ r ≤ b, the honeycomb of the photonic crystal structure b ≤ r ≤ c, and finally the outer silica cladding ≤ c ≤ r ≤ d.

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The generalized Hooke’s Law expression [18], relating the independent stresses and strains for an orthotropic material, assuming there is no shearing within the material, is

[\begin{matrix} ϵ_{11} \\ ϵ_{22} \\ ϵ_{33} \end{matrix}] = [\begin{matrix} \frac{1}{E_{1}} & \frac{- ν_{21}}{E_{2}} & \frac{- ν_{31}}{E_{3}} \\ \frac{- ν_{12}}{E_{1}} & \frac{1}{E_{2}} & \frac{- ν_{32}}{E_{3}} \\ \frac{- ν_{13}}{E_{1}} & \frac{- ν_{23}}{E_{2}} & \frac{1}{E_{3}} \end{matrix}] [\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \end{matrix}],

where E_j are the elastic moduli in their respective directions, and ν_ij are the materials Poisson’s ratios between the i and j directions, and we note that the indices 11, 22 and 33 correspond to the radial, azimuthal, and longitudinal directions in the fiber; r, θ and z.

The hollow core photonic crystal fiber layers have two distinct symmetries: isotropic and transversely isotropic. To solve for the longitudinal strain in the fiber the stress-strain relationships for these two symmetries will be needed. The photonic crystal, a honeycomb structure, behaves mechanically as a transversely isotropic material, different from its constituent material, fused (isotropic) silica. The core-sheath and cladding, consisting of solid fused silica thus has E_j = E_S, the Young’s Modulus for silica, and ν_ij = ν_S, the Poisson’s ratio for silica. So then for these solid layers

[\begin{matrix} ϵ_{r} \\ ϵ_{θ} \\ ϵ_{z} \end{matrix}] = [\begin{matrix} \frac{1}{E_{S}} & \frac{- ν_{S}}{E_{S}} & \frac{- ν_{S}}{E_{S}} \\ \frac{- ν_{S}}{E_{S}} & \frac{1}{E_{S}} & \frac{- ν_{S}}{E_{S}} \\ \frac{- ν_{S}}{E_{S}} & \frac{- ν_{S}}{E_{S}} & \frac{1}{E_{S}} \end{matrix}] [\begin{matrix} σ_{r} \\ σ_{θ} \\ σ_{z} \end{matrix}] .

By inverting Eq. (10) we have

[\begin{matrix} σ_{r} \\ σ_{θ} \\ σ_{z} \end{matrix}] = [\begin{matrix} (λ + 2 μ) & λ & λ \\ λ & (λ + 2 μ) & λ \\ λ & λ & (λ + 2 μ) \end{matrix}] [\begin{matrix} ϵ_{r} \\ ϵ_{θ} \\ ϵ_{z} \end{matrix}],

where

λ = \frac{E_{S} ν_{S}}{(1 + ν_{S}) (1 - 2 ν_{S})}

and

μ = \frac{E_{S}}{2 (1 - 2 ν_{S})}

are the so called Lamé parameters. We will need similar expressions for the photonic crystal structure, however, the honeycomb-like structure of the photonic crystal as mentioned above acts as a transversely isotropic material–a material that is rotationally symmetric about one axis–and is slightly different for the regular hexagon and kagome crystal structures.

4.1. Hexagon honeycomb

The honeycomb made from regular hexagons in a trigonal pattern as seen in Fig. 4 has a relative density of $ρ^{h e x} / ρ_{S} = 2 t / \sqrt{3 l}$ , where ρ_S is the density of silica, t is the honeycomb wall thickness, and l is the cell wall length [18]. The relative density is used to define the air-filling ratio by ρ/ρ_S ≡ (1 − η). The transverse and longitudinal elastic moduli and Poisson’s ratios are given in Table 3. In this structure the elastic moduli in the transverse direction are $E_{1} = E_{2} = E_{T}^{h e x}$ and in the longitudinal direction $E_{3} = E_{L}^{h e x}$ , and the Poisson’s ratios are ν_T = 1, ν_L = ν_s. Then Hooke’s law (ignoring shear) for this regular hexagon honeycomb is

[\begin{matrix} ϵ_{r} \\ ϵ_{θ} \\ ϵ_{z} \end{matrix}] = [\begin{matrix} \frac{1}{E_{T}^{h e x}} & \frac{- 1}{E_{T}^{h e x}} & \frac{- ν_{S}}{E_{L}^{h e x}} \\ \frac{- 1}{E_{T}^{h e x}} & \frac{1}{E_{T}^{h e x}} & \frac{- ν_{S}}{E_{L}^{h e x}} \\ \frac{- ν_{S}}{E_{L}^{h e x}} & \frac{- ν_{S}}{E_{L}^{h e x}} & \frac{1}{E_{L}^{h e x}} \end{matrix}] [\begin{matrix} σ_{r} \\ σ_{θ} \\ σ_{z} \end{matrix}] .

Fig. 4 A regular hexagonal honeycomb crystalline structure. This photonic crystal is made from close-packed regular hexagons whose side length is l and whose side thickness is t. The hexagon edges in this case represent the silica material.

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Table 3. Elastic properties of two crystal-like structures and fused silica in terms of the corresponding property in fused silica, see [18–20]. The air filling ratio η relates to the cell wall thickness, t, and wall length, l, by $η = (1 - 2 t / \sqrt{3 l})$ . Crystal structures are as shown in Fig. 4–5.

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Inverting the above Eq. (12) we find

[\begin{matrix} σ_{r} \\ σ_{θ} \\ σ_{z} \end{matrix}] = ψ^{h e x} [\begin{matrix} β^{h e x} & γ^{h e x} & δ^{h e x} \\ γ^{h e x} & β^{h e x} & δ^{h e x} \\ δ^{h e x} & δ^{h e x} & 0 \end{matrix}] [\begin{matrix} ϵ_{r} \\ ϵ_{θ} \\ ϵ_{z} \end{matrix}],

where

ψ^{h e x} = \frac{(1 - η) E_{s}}{8 ν_{s}^{2}}

,

β^{h e x} = (3 {(1 - η)}^{2} ν_{s}^{2} - 2)

,

γ^{h e x} = (- 3 {(1 - η)}^{2} ν_{s}^{2} - 2)

and δ^hex = (−4ν_s). The cubic dependency of the transverse moduli is of particular interest. Also we note that the forms of the above equation and Eq. (12) preserve the crystal symmetry.

We should make a special mention about the relationships ν₂₃/E₂ = ν₃₂/E₃ and ν₁₃ E₁ = ν₃₁/E₃ for the hexagon honeycomb. In the literature, there are some instances where ν₂₃ = ν₃₂E₂/E₃ = (1−η)²ν_S → 0, since for practical purposes η → 1 for a high air-filling ratio [5,17,21]. However, this choice presents a problem as it makes the determinant of the matrix in Eq. (12) zero, which in turn makes the inverse of that matrix, in Eq. (13) undefined. To resolve the issue we rely on the crystal symmetry of the matrix.

4.2. Kagome lattice honeycomb

Another type of honeycomb, made from equilateral triangles and regular hexagons, in a kagome pattern as seen in Fig. 5 has a relative density of $ρ^{k a g} / ρ_{S} = \sqrt{3} t / l$ or in terms of the air-filling ratio defined before (3/2)(1 − η). The transverse and longitudinal elastic moduli follow the same relationships as for the regular hexagon honeycomb, and their values are given in Table 3, along with the related Poisson’s ratios. Now Hooke’s law (ignoring shear) for this kagome lattice honeycomb is

[\begin{matrix} ϵ_{r} \\ ϵ_{θ} \\ ϵ_{z} \end{matrix}] = [\begin{matrix} \frac{1}{E_{T}^{k a g}} & \frac{- 1}{3 E_{T}^{k a g}} & \frac{- ν_{S}}{E_{L}^{k a g}} \\ \frac{- 1}{3 E_{T}^{k a g}} & \frac{1}{E_{T}^{k a g}} & \frac{- ν_{S}}{E_{L}^{k a g}} \\ \frac{- ν_{S}}{E_{L}^{k a g}} & \frac{- ν_{S}}{E_{L}^{k a g}} & \frac{1}{E_{L}^{k a g}} \end{matrix}] [\begin{matrix} σ_{r} \\ σ_{θ} \\ σ_{z} \end{matrix}] .

Fig. 5 A hexagonal kagome crystalline structure. This photonic crystal is made from equilateral triangles and regular hexagons whose side length are l and whose side thickness are t.

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Inverting the above Eq. (14) we find

[\begin{matrix} σ_{r} \\ σ_{θ} \\ σ_{z} \end{matrix}] = ψ^{k a g} [\begin{matrix} β^{k a g} & γ^{k a g} & δ^{k a g} \\ γ^{k a g} & β^{k a g} & δ^{k a g} \\ δ^{k a g} & δ^{k a g} & 1 \end{matrix}] [\begin{matrix} ϵ_{r} \\ ϵ_{θ} \\ ϵ_{z} \end{matrix}],

where for

ψ^{k a g} = \frac{3 (1 - η) E_{s}}{2 (1 - ν_{s}^{2})}

,

β^{k a g} = (\frac{3 - ν_{s}^{2}}{8})

, and

γ^{k a g} = (\frac{1 + ν_{s}^{2}}{8})

, and

δ^{k a g} = (\frac{ν_{s}}{2})

. Again, Eq. (14) and (15) preserve the transverse isotropic crystal symmetry–with one noticeable difference to the two similar expressions for the regular honeycomb. In the lower right-hand corner of Eq. (15) the 0 → 1 as a result of the different transverse Poisson’s ratio.

4.3. Temperature dependence

To introduce temperature dependence in the strains in the above equations, the coefficient(s) of thermal expansion are included, and the original strains are replaced by

[\begin{matrix} ϵ_{r} \\ ϵ_{θ} \\ ϵ_{z} \end{matrix}] = [\begin{matrix} ϵ_{r}^{'} \\ ϵ_{θ}^{'} \\ ϵ_{z}^{'} \end{matrix}] - [\begin{matrix} α_{r} \\ α_{θ} \\ α_{z} \end{matrix}] Δ T

where

ϵ_{r}^{'}

are the temperature independent strains (often referred to as mechanical strains), ΔT is a change in temperature, and α_i are the coefficients of thermal expansion (CTE), most generally a tensor, see [22]. The CTE is independent of direction for isotropic materials. For solid fused silica the linear CTE

α_{L}^{s i l i c a} = 0.55

ppm/K.

In the case of the photonic crystal structures the CTE should also reflect the crystal symmetry, the CTE in the longitudinal direction should be different from the CTE transverse directions, an α_L and an α_T, respectively. Unfortunately, not much work exists in the literature regarding the specific relationship these coefficients have on the exact crystal geometry, i.e. with respect to the air-filling ratio. Work was done by Barker, [23], exploring the relationship between elastic moduli and CTEs, and more recently by others relating them to Poisson’s ratios and Grüneisen parameters, [24, 25], and others. We assume Barker’s Rule, Eα² ∝ const for i = r, θ, z, namely $E_{i} α_{i}^{2} \propto A E_{S} α_{S}^{2}$ , where A is a constant, as a possible approximation for finding the CTEs of the two crystal structures, another limitation being the nominal linear CTE of the constituent material, in this case fused silica. It is reasonable to assume that as the air-filling ratio increases, the stiffness of the crystal lowers, and thus permits greater thermal expansion. The overall expansion would be limited by the rate at which the cladding layer expands as it is more stiff than, and encompasses, the crystal layer.

While we adopt Barker’s Rule here, it has been convention to assume that if the cellular material is isotropic in the plane (as the above crystal structures are) then the CTE should be that of the constituent material.

4.4. Solving the longitudinal strain

Appropriate boundary conditions are necessary to be able to solve for the Lamé solutions, constants K_1,_i, K_2,_i, and K_3,_i used to define the strain for the ith layer in the fiber.

From the literature, see [8, 13, 16, 26] etc., for the ith layer, the radial, azimuthal, and longitudinal strains – without thermal dependence – are given by the solutions:

ϵ_{r, i}^{'} = K_{1, i} + \frac{K_{2, i}}{r^{2}},

ϵ_{θ, i}^{'} = K_{1, i} - \frac{K_{2, i}}{r^{2}},

ϵ_{z, i}^{'} = K_{3, i} .

For any plane in the fiber perpendicular to the fiber’s length, far enough away from the fiber’s ends, we also expect the sum of forces in the longitudinal direction to be zero. Similarly at any point within the fiber the sum of forces in the radial direction must be zero, as a condition of mechanical equilibrium. At any boundary between layers we hold that the radial displacement of the boundary is identical as viewed from either side of it. As a condition brought on by assuming no internal shearing: one layer stretching more or less than the neighboring layer, across the fiber’s cross-section is forbidden.

It has been convention to take

ϵ_{z, i}^{'} = ϵ_{z, 2}^{'} = ϵ_{z, 3}^{'} = K_{3},

which we perceive to be a misstep. It should be the case that at a given point along the fiber length, if there is no internal shearing, that the longitudinal displacement is the same for any radius within the cross section of the fiber, and trivially that those displacements are continuous. We can’t make statements like

{ϵ_{z, i} |}_{r} = {ϵ_{z, j} |}_{r}

(the derivative of the displacement) for any of the layers we consider here without including the thermal expansion terms, α_iΔT. So we take

ϵ_{z, 1} \Rightarrow K_{3, 1} - α_{z, 1} Δ T,

ϵ_{z, 2} \Rightarrow K_{3, 2} - α_{z, 2} Δ T,

ϵ_{z, 3} \Rightarrow K_{3, 3} - α_{z, 3} Δ T,

and state

ϵ_{z, 1} = ϵ_{z, 2} = ϵ_{z, 3} .

Newton’s second law for the longitudinal direction follows

\sum {\vec{F}}_{z} = 0 = \sum_{i} σ_{z, i} A_{i} \leftrightarrow \int_{0}^{2 π} \int_{0}^{R} σ_{z} r d θ d r

where

A_{i} = π (r_{o u t, i}^{2} - r_{i n, i}^{2})

is the annular area of the ith layer with r_out and r_in as the outer and inner radii of that layer, or via the integration with R as the outer radius of the fiber. This does ignore end effects which is reasonable in the case of a very long, thin fiber.

Considering the sum of forces in the radial direction $\sum {\vec{F}}_{r} = 0$ , the stress acting from exterior to, and the stress acting from interior of a cylindrical surface element must be equal to each other. Thus:

{σ_{r, 1} |}_{r = a} = - P_{1},

{σ_{r, 1} |}_{r = b} = {σ_{r, 2} |}_{r = b},

{σ_{r, 2} |}_{r = c} = {σ_{r, 3} |}_{r = c},

{σ_{r, 3} |}_{r = d} = - P_{2},

for the surfaces where the layers touch, and for where they meet the hollow core or external gas pressures. This is a deviation from [8] where Eq. (23) and (26) are equal to zero. The last boundary conditions we have are the radial displacement equations. They are found from

u_{r, i} = \int ϵ_{r, i} d r

where we make certain we mean the temperature dependent strain. So then the final conditions are

{u_{r, 1} |}_{r = b} = {u_{r, 2} |}_{r = b},

{u_{r, 2} |}_{r = c} = {u_{r, 3} |}_{r = c} .

With the nine boundary conditions, Eq. (21–28) it is possible to analytically solve the system for K_1,_i, K_2,_i and K_3,_i. The response we’re interested in ϵ_z_,2 = S_L ΔT, if the radial temperature dependence ΔT is simple enough. We take the dependence to be be constant across the fiber cross section for a quasi-static scenario where the temperature change is slower than heat diffusion in the fiber.

A straight-forward approach of constructing the system of equations in a matrix format and inputting appropriate numerical values is implemented in MATLAB to solve for the unknown constants. Upon reducing the thickness of the core-sheath layer to zero, we find that the three layered system of equations approaches that of a two layered system (without the core-sheath layer), as would be expected, and that zero thickness leads to perfect agreement. Since in many commercially available hollow core fibers the thickness of the (ideally) cylindrical ring of material surrounding the core is as thin as the photonic crystal cell walls, it may be considered part of the crystal layer. Without removing the core-sheath layer, where its thickness is the same as the photonic crystal cell wall thickness, the difference between the calculated ϵ_z_,2 for the three and two layer solution, for example in an ideal HC-1550-02 fiber, is 0.6%.

Since a real photonic crystal fiber does not have a perfect crystal structure truncated at definite radii as described above, the numerical solution of S_L as shown in Table 2 is a quick, coarse guide used to guide our analysis. Other works (see for instance the full approach used in [8]) have made pointed efforts for a more accurate simulation of S_L. To precisely simulate the thermal phase response due to change of length of a particular fiber a more computationally intensive method using finite element analysis or an equivalent technique would be warranted to account for deviations from the regular crystal structure.

For comparison, the core, holey region, and outer cladding diameters for NKT Photonics’ HC19-1550 are 20 ± 2µm, 70 ± 5µm and 115 ± 3µm respectively [27]. The core, holey region, and outer cladding diameters for NKT Photonics’ HC-1550-02 are 10 ± 1µm, 70 ± 5µm and 120 ± 2µm respectively [28]. The core, holey region, and outer cladding diameters for GLOphotonics’ PMC-C-TiSa-7C are 63 ± 1µm, 180 ± 1µm and 300 ± 1µm respectively [29]. For both of the NKT fibers we estimate that the cellular wall thickness was around 0.06µm, while the structure wall length was around 1µm. For the GLOphotonics fiber we estimated that the cell wall thickness was also around 0.06µm and the structure wall length was about 6µm. These estimates were made from observation of the technical specification literature made available online by both companies, and from SEM images we have taken of the fiber cross sections. The estimated values of the cell wall thickness may not be accurate and are only reported as approximate. There are no nominal values reported by their respective manufacturers. Additionally, we modeled the solid core SMF-28 fiber which has a core diameter of 8.2µm and a cladding diameter of 125 ± 0.7µm [30].

For the numerical results we took for the value of Young’s modulus of fused silica to be E_S = 73 GPa, the Poisson’s ratio as ν_S = 0.17, the linear coefficient of thermal expansion as α_S = 0.55e-6 K⁻¹, and the internal and external pressures to be one atmosphere (0.101325 MPa). Assuming a vacuum in the interior of the fiber changes the results by less than 5% for the photonic bandgap fibers, and by less than 0.5% for the kagome lattice fiber.

Values for the numerical result of the two layer solution for the HC19-1550, HC-1550-02, or PMC-C-TiSa_Er-7C fibers, and for SMF-28 are in Table 2. We conclude from them that the kagome lattice structure does have a higher thermal phase sensitivity than that of the regular hexagon photonic crystal structures and correlate this with the increased stiffness in the transverse direction. From the numerical results for the HC19-1550 and HC-1550-02 fibers, we believe that the lower S_L value of the two is due to the larger core size and acknowledge that the accompanying experimental results do not clearly favor either fiber as having the lowest thermal phase sensitivity.

5. Conclusions

We have found an explanation that satisfactorily resolves the seeming discrepancy in the literature regarding the thermal phase sensitivity of hollow core photonic crystal fibers presented by [7]. Furthermore, we tested two similiar hollow core fibers to compare with [7, 8] and found general agreement with those results. Additionally, we report the first known testing and modeling the thermal phase sensitivity in an inhibited coupling hollow core fiber with a kagome lattice photonic crystal structure. By not including a inner-most silica core-sheath and outer jacketing layers in the model and assuming the coefficient of thermal expansion for the photonic crystal layer had a longitudinal and transverse component, we found numerical results that closely match experimental data.

The analytical model used by Dangui, et. al works well for ideal transversely isotropic photonic crystal structures, these include photonic bandgap and inhibited coupling guiding mechanisms - as they are well described mechanical systems. Recently a new form of air-guiding fibers, antiresonant guiding fibers [31, 32], have taken shape that have interesting mechanical and possible thermal characteristics, however their geometry prevents the implementation of this mechanical model to numerically simulate thermal phase sensitivity due to change in length and strain. As these fibers do not have regular crystal lattice structures the approach of Dangui, et. al is insufficient; some other means, such as a finite element method, would be necessary to calculate the thermo-mechanically induced fiber strain and thus the thermal phase response of this kind of fiber.

There are however some subtle ambiguities that still exist regarding the simulation of the thermal phase response in a hollow core photonic crystal fiber, some of which have been mentioned in the literature. As addressed earlier, appropriate symmetry of the stress-strain relationship is important to preserve. The appropriate boundary conditions are also not entirely consistent in the mechanical models of [15,26,33] and in [8,13] as well as in [5,17,21]. Where the air-core and the photonic crystal interface, and where the cladding material meets and external gas at some non-zero pressure, the radial stress boundary conditions must reflect the internal and external pressures on the material. The crystal structure defects directly surrounding the core, and at the interior of the cladding layer, influence the ideal modeling as described in this paper, and by others, and may impact the thermal phase response of hollow core photonic crystal fibers to a significant degree. Possibly the most subtle of these issues is the exact relationship the transverse and longitudinal linear coefficients of thermal expansion have with each other and with the corresponding elastic moduli. While in this paper we’ve taken a novel approach by assuming Barker’s Rule for the CTE in the numerical modeling, this is generally not the assumption made. The heart of this issue is outside the scope of this communication, and we leave it for future investigation.

Fiber history

Blaze Photonics was acquired by Crystal Fibre in 2004. Crystal Fibre became NKT Photonics in 2009. It should be expected that results for the Blaze Photonics’ HC-1550-02 fiber would be at the least nearly identical to NKT Photonics’ HC-1550-02 fiber results. Differences can be attributed to small variations in the fiber manufacturing process, and from the length of fiber whose properties are measured.

Funding

U.S. Naval Research Laboratory Base Program for 6.1 Basic Research.

References and links

1. H. Lefévre, “Fundamentals of the interferometric fiber-optic gyroscope,” Optical Review 4, 20–27 (1997). [CrossRef]

2. J. A. Bucaro, N Lagakos, J. H. Cole, and T. G. Giallorenzi, “Fiber optic acoustic transduction,” in Physical Acoustics Vol. XVI: Principles and Methods, W. P. Mason and R. N. Thurston, eds. (Academic, 1982). [CrossRef]

3. R. Cregan, B. Mangan, J. Knight, T. Birks, P. Russell, P. Roberts, and D. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]

4. S. Blin, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Reduced thermal sensitivity of a fiber-optic gyroscope using an air-core photonic-bandgap fiber,” J. Lightwave Technol. 25, 861–865 (2007). [CrossRef]

5. M. Pang and W. Jin, “Detection of acoustic pressure with hollow-core photonic bandgap fiber,” Opt. Express 17, 11088–11097 (2009). [CrossRef] [PubMed]

6. G. A. Cranch and G. A. Miller, “Coherent light transmission properties of comercial photonic crystal hollow core optical fiber,” Appl. Opt. 54, F8–F16 (2015). [CrossRef] [PubMed]

7. R. Slavik, G. Marra, E. Numkam Fokoua, N. Baddela, N. V. Wheeler, M. Petrovich, F. Poletti, and D. J. Richardson, “Ultralow thermal sensitivity of phase and propagation delay in hollow core optical fibres,” Sci. Rep. 5, 15447 (2015). [CrossRef] [PubMed]

8. V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, “Phase sensitivity to temperature of the fundamental mode in air-guiding photonic-bandgap fibers,” Opt. Express 13, 6679–6684 (2005). [CrossRef]

9. F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. Numkam Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavik, and D. J. Richardson, “Towards high-capacity fibre-optic communications at the speed of light in vacuum,” Nature Photonics 7, 279–284 (2013). [CrossRef]

10. A. H. Hartog, A. J. Conduit, and D. N. Payne, “Variation of pulse delay with stress and temperature in jacketed and unjacketed optical fibres,” Optical and Quantum Electronics 11, 265–273 (1978). [CrossRef]

11. R. Kashyap, S. Hornung, M. H. Reeve, and S. A. Cassidy, “Temperature desensitisation of delay in optical fibres for sensor applications,” Electron. Lett. 19, 1039–1040 (1983). [CrossRef]

12. M. Tateda, S. Tanaka, and Y. Sugawara, “Thermal characteristics of phase shift in jacketed optical fibers,” Appl. Opt. 19, 770–773 (1980). [CrossRef] [PubMed]

13. N. Lagakos, J. A. Bucaro, and J. Jarzynski, “Termperature-induced optical phase shifts in fibers,” Appl. Opt. 20, 2305–2308 (1981). [CrossRef] [PubMed]

14. B. Budiansky, D. C. Drucker, G. S. Kino, and J. R. Rice, “Pressure sensitivity of a clad optical fiber,” Appl. Opt. 18, 4085–4088 (1979). [CrossRef] [PubMed]

15. R. Hughes and J. Jarzynski, “Static pressure sensitivity amplification in interferometric fiber-optic hydrophones,” Appl. Opt. 19, 98–107 (1980). [CrossRef] [PubMed]

16. S. W. Løvseth, J. T. Kringlebotn, E. Rønnekliev, and K. Bløtekær, “Fiber distributed-feedback lasers used as acoustic sensors in air,” Appl. Opt. 38, 4821–4830 (1999). [CrossRef]

17. M. Pang, H. F. Xuan, J. Ju, and W. Jin, “Influence of strain and pressure to the effective refractive index of the fundamental mode of hollow-core photonic bandgap fibers,” Opt. Express 18, 14041–14055 (2010). [CrossRef] [PubMed]

18. L. J. Gibson and M. F. Ashby, Cellular Solids: Structures and Properties, 2nd ed. (Cambridge University, 1997). [CrossRef]

19. B. Kim and R. M. Christensen, “Basic two-dimension core types for sandwich structures,” International Journal of Mechanical Sciences 42, 657–676 (2000). [CrossRef]

20. A.-J. Wang and D. L. McDowell, “In-plane stiffness and yield strength of periodic metal honeycombs,” Journal of Engineering Materials and Technology 126, 137–156 (2004). [CrossRef]

21. Y. Cao, W. Jin, F. Yang, and H. L. Ho, “Phase sensitivity of fundamental mode of hollow-core photonic bandgap fiber to internal gas pressure,” Opt. Express 22, 13190–13201 (2014). [CrossRef] [PubMed]

22. R. F. Tinder, Tensor Properties of Solids (Morgan & Claypool, 2008).

23. R. E. Barker Jr., “An approximate relation between elastic moduli and thermal expansivities,” J. Appl. Phys. 34, 107–116 (1963). [CrossRef]

24. D. S. Sanditov and B. S. Sydykon, “Modulus of elasticity and thermal expansion coefficient of glassy solids,” Physics of the Solid State 56, 1006–1008 (2014). [CrossRef]

25. D. S. Sanditov and M. V. Darmaev, “Elactic moduli and poisson’s coeficient of optical glasses,” Russian Physics Journal 58, 683–690 (2015). [CrossRef]

26. G. B. Hocker, “Fiber-optic sensing of pressure and temperature,” Appl. Opt. 18, 1445–1448 (1979). [CrossRef] [PubMed]

27. NKT Photonics Inc., Technical Specifications for HC19-1550, 1400 Campus Drive West, Morganville, NJ 07751, USA (2016).

28. NKT Photonics Inc., Technical Specifications for HC-1550-02, 1400 Campus Drive West, Morganville, NJ 07751, USA (2016).

29. GLOphotonics, Technical Specifications for PMC-C-TiSa_Er-7C, 1 Avenue d’Ester, 87069 Limoges Cedex, France (2016).

30. Corning, Technical Specifications for SMF-28, One Riverfront Plaza, Corning, NY 14831 USA (2016).

31. A. Argyros, S. G. Leon-Saval, J. Pla, and A. Docherty, “Antiresonant reflection and inhibited coupling in hollow-core square lattice optical fibers,” Opt. Express 16, 5642–5648 (2008). [CrossRef] [PubMed]

32. F. Poletti, “Nested antiresonant nodeless hollow core fiber,” Opt. Express 22, 23807–23828 (2014). [CrossRef] [PubMed]

33. G. B. Hocker, “Fiber optic acoustic sensors with composite structure: an analysis,” Appl. Opt. 18, 3679–3683 (1979). [CrossRef] [PubMed]

	HC-PBGF [7]	Corning SMF-28e+ [7]	Crystal Fibre AIR-10-1550 [8]	Blaze Photonics HC-1550-02 [8]	Corning SMF-28 [8]

Length L	10.6	2.1	2.26	2.1	2.1
Meas. S_ϕ	(0.6)	(7.7)	1.5±0.9	2.2±0.7	7.9
$Meas . \frac{1}{L} \frac{d τ}{d T}$	2	37.4	(5±3)	(7±2)	(38.2)

Type	NKT Photonics HC19-1550 Hexagon	NKT Photonics HC-1550-02 Hexagon	GLOphotonics PMC-C-TiSa_Er-7C Kagome	Corning SMF-28 Solid core

Length L	9.30	8.15	3.31	10.63
Model S_L	0.91	0.92	1.33	0.65
Meas. S_ϕ	0.95 ± 0.05	0.88 ± 0.05	1.21 ± 0.10	6.88 ± 0.39
$\frac{1}{L} \frac{d τ}{d T}$	(3.2 ± 0.2)	(2.9 ± 0.2)	(4.0 ± 0.3)	(35 ± 2)

	HC-PBGF [7]	Corning SMF-28e+ [7]	Crystal Fibre AIR-10-1550 [8]	Blaze Photonics HC-1550-02 [8]	Corning SMF-28 [8]

Length L	10.6	2.1	2.26	2.1	2.1
Meas. S_ϕ	(0.6)	(7.7)	1.5±0.9	2.2±0.7	7.9
$Meas . \frac{1}{L} \frac{d τ}{d T}$	2	37.4	(5±3)	(7±2)	(38.2)

Type	NKT Photonics HC19-1550 Hexagon	NKT Photonics HC-1550-02 Hexagon	GLOphotonics PMC-C-TiSa_Er-7C Kagome	Corning SMF-28 Solid core

Length L	9.30	8.15	3.31	10.63
Model S_L	0.91	0.92	1.33	0.65
Meas. S_ϕ	0.95 ± 0.05	0.88 ± 0.05	1.21 ± 0.10	6.88 ± 0.39
$\frac{1}{L} \frac{d τ}{d T}$	(3.2 ± 0.2)	(2.9 ± 0.2)	(4.0 ± 0.3)	(35 ± 2)

Optical phase response to temperature in a hollow-core photonic crystal fiber

Abstract

1. Introduction

2. Definition of sensitivity

3. Experiment

4. Thermo-mechanical sensitivity

4.1. Hexagon honeycomb

4.2. Kagome lattice honeycomb

4.3. Temperature dependence

4.4. Solving the longitudinal strain

5. Conclusions

Fiber history

Funding

References and links

Cited By

Figures (5)

Tables (3)

Equations (32)

Optics Express

Elastic Moduli & Poisson Ratios
	$\frac{ρ}{ρ_{S}}$	$\frac{E_{T}}{E_{S}}$	ν_T	$\frac{E_{L}}{E_{S}}$	ν_L
Silica	1	1	ν_S	1	ν_S
Hexagon	(1 − η)	$\frac{3}{2} {(1 - η)}^{3}$	1	(1 − η)	ν_S
Kagome	$\frac{3}{2} (1 - η)$	$\frac{1}{2} (1 - η)$	1	$\frac{3}{2} (1 - η)$	ν_S