## Abstract

Because in an air-core photonic-bandgap fiber the fundamental mode travels mostly in air, as opposed to silica in a conventional fiber, the phase of this mode is expected to have a much lower dependence on temperature than in a conventional fiber. We confirm with interferometric measurements in air-core fibers from two manufacturers that their thermal phase sensitivity is indeed ~3 to ~6 times smaller than in an SMF28 fiber, in agreement with an advanced theoretical model. With straightforward fiber design changes (thinner jacket and thicker outer cladding), this sensitivity could be further reduced down to ~11 times that of a standard fiber. This feature is anticipated to have important benefits in fiber optic systems and sensors, especially in the fiber optic gyroscope where it translates into a lower Shupe effect and thus a greater long-term stability.

©2005 Optical Society of America

## 1. Introduction

The optical phase of a signal traveling in an optical fiber is a relatively strong function of temperature. As the temperature changes, the fiber length, radius, and refractive indices all change, which results in a change in the signal phase. This effect is generally sizable and detrimental in phase-sensitive fiber systems such as fiber sensors utilizing conventional fibers. For example, in a fiber sensor based on a Mach-Zehnder interferometer with 1-m long arms, a temperature change in one of the arms as small as 0.01°C is sufficient to induce a differential phase change between the two arms as large as ~1 rad. This is about a million times larger than the typical minimum detectable phase of an interferometric sensor (~1 μrad). Taking care of this large phase drift is often a significant challenge. A particularly important sensor where thermal effects have been troublesome is the fiber optic gyroscope (FOG). Although the FOG utilizes an inherently reciprocal Sagnac interferometer, even a small asymmetric change in the temperature distribution of the Sagnac coil fiber will result in a differential phase change between the two counter-propagating signals, a deleterious effect known as the Shupe effect.[1,2] Because the Sagnac is a common-path interferometer, the two signals see almost the same thermally induced change and this differential phase change is much smaller than in a Mach-Zehnder or Michelson interferometer, but it is not small enough for high-accuracy applications, which require extreme phase stability. In this and other fiber sensors and systems, thermal effects have been successfully fought with clever engineering solutions. These solutions, however, generally increase the complexity and cost of the final product, and they can also negatively impact its reliability and lifetime.

These undesirable thermal effects are anticipated to be greatly reduced by replacing the conventional fiber with an air-core photonic-bandgap fiber (PBF). The reason is that in a PBF, most of the mode energy is confined in air, unlike in a conventional fiber where the mode travels entirely through silica. Since the thermal coefficient of the refractive index *dn*/*dT* is much smaller for air than for silica, in an air-core fiber the temperature sensitivity of the mode effective index is expected to be reduced considerably. The length of a PBF of course still varies with temperature, which means that the phase sensitivity will not be reduced simply in proportion to the percentage of mode energy in silica. However, it should still be reduced significantly, an improvement beneficial to numerous applications, especially in the FOG where it implies a reduced Shupe effect.

The objectives of this paper are first to model quantitatively the dependence of the fundamental-mode phase on temperature in an air-core fiber, second to validate these predictions by comparing them to values measured in actual air-core fibers, and third to design improved fibers with even lower temperature sensitivity. The metric in this work is the relative change in phase *S*, which we refer to as the phase thermal constant:

where *ϕ* is the phase accumulated by the fundamental mode through the fiber and *T* is the fiber temperature. With an interferometric technique, we tested two air-core PBFs from different manufacturers and found that their thermal constant is in the range of 1.5 to 3.2 parts per million (ppm) per degree Celsius, or 2.5–5.2 times lower than the measured value of a conventional SMF28 fiber (*S* = 7.9 ppm/°C). Each of these values falls within 20% of the corresponding predicted number, which lends credence to our theoretical model and to our measurement calibration. This study shows that the reason for this reduction is due to a drastic reduction in the dependence of the mode effective index on temperature. The residual value of the thermal constant arises from length expansion of the fiber, which is only marginally reduced in an air-core fiber. We show through modeling that with straightforward fiber jacket improvements, this contribution can be further reduced by a factor of ~2. This study establishes even without this further improvement, the phase thermal constant of current air-core fiber is as much as ~5 times smaller than in a conventional fiber, which should bring forth a significant improvement in the FOG and other phase-sensitive systems.

## 2. Model of thermal effects in air-core fibers

In this section, we develop a theoretical model to quantify the phase thermal constant *S* of an air-guided photonic-bandgap fiber and, for comparison purposes, of a conventional index-guided fiber as well. The phase *ϕ* accumulated by the fundamental mode as it propagates through a fiber of length *L* is:

where *L* is the fiber length, *n _{eff}* the mode effective index, and

*λ*the wavelength of the signal in vacuum. Inserting Eq. (2) in Eq. (1) yields the expression for the phase sensitivity per unit length and per degree of temperature change of the fiber:

*S* is the sum of two terms: the relative variation in fiber length per degree of temperature change (hereafter called *S _{L}*), and the relative variation in the mode effective index per degree of temperature change (hereafter called

*S*).

_{n}If the temperature change from equilibrium is Δ*T*(*t*,*l*) at time t and in an element of fiber length *dl* located a distance *l* from one end of the fiber, the total phase change in a length *L* of the fiber is:

where *v* = *c*/*n _{eff}* and

*c*is the velocity of light in vacuum. Equation (4) shows that

*S*is a relevant parameter to characterize the phase sensitivity to temperature in, for instance, a Mach-Zehnder interferometer, since the total phase change is proportional to

*S*.

Similar expressions apply to other interferometers. For example, for a Sagnac interferometer the corresponding phase change is given by:

As expected, it is proportional to *S*, and *S* is again the relevant metric. The temperature sensitivity of a Sagnac interferometer (Eq. (5)) can be reduced by minimizing the integral through proper fiber winding,[3,4] as is well known, and/or by designing the fiber structure to minimize *S*. This work is concerned only with this second technique.

Because the thermal expansion coefficient of the fiber jacket (usually a polymer) is typically two orders of magnitude larger than that of silica, expansion of the jacket stretches the fiber, and the fiber length change caused by jacket expansion is the dominant contribution to *S _{L}*. The index term

*S*is the sum of three effects. The first one is the transverse thermal expansion of the fiber, which modifies the core radius and the photonic-crystal dimensions, and thus the mode effective index. The second effect is the strains that develop in the fiber as a result of thermal expansion; these strains alter the effective index through the elasto-optic effect. The third effect is the change in material indices induced by the fiber temperature change (thermo-optic effect).

_{n}#### 2.1 Exact model

To determine *S _{n}* and

*S*, we need to model the thermo-mechanical properties of the fiber. To do so, we assume that the fiber temperature is changed uniformly from

_{L}*T*to

_{0}*T*+

_{0}*dT*, and we aim to calculate the fiber length and the effective index of the fundamental mode at both temperatures, from which, with Eq. (3), we can calculate

*S*,

_{n}*S*, and

_{L}*S*. We assume that the fiber has cylindrical symmetry and that all its properties are invariant along its length, so we can compute

*S*in a cylindrical coordinate system. As shown in Fig. 1, the fiber is modeled as a structure with multiple circular layers: a core (doped silica in a conventional fiber, air in a PBF) of radius

*a*

_{0}, an inner cladding (silica in a conventional fiber, a silica–air honeycomb in a PBF), an outer cladding (generally pure silica) of radius

*a*, and a jacket (often an acrylate). We assume that each layer remains in contact and in mechanical equilibrium with the neighboring layers, i.e., the radial stress and the radial deformation are continuous across fiber layer boundaries.

_{M}Each layer is characterized by a certain elastic modulus *E*, Poisson’s ratio *ν*, and thermal expansion coefficient *α*. The photonic-crystal cladding is an exception in that it is not a homogeneous material but it behaves mechanically like a honeycomb. The implications are that (1) in a transverse direction the honeycomb can be squeezed much more easily than a solid, which means that it has a high transverse Poisson’s ratio, and (2) in the longitudinal direction, it behaves like membranes of silica with a total area (1-*η*)*A _{h}*, where

*A*is the total cross section of the honeycomb. The elastic modulus and Poisson’s ratio of a honeycomb are thus function of the air filling ratio

_{h}*η*. For an hexagonal pattern of air holes in silica, they are given by: [5]

where *E _{T}* and

*E*are the transverse and longitudinal Young’s modulus of the silica-air honeycomb, respectively,

_{L}*E*is the Young’s modulus of silica,

_{0}*ν*and

_{T}*ν*are the transverse and longitudinal Poisson’s ratios of the honeycomb material, respectively, and

_{L}*ν*is the Poisson’s ratio of silica. The values used for these parameters in our simulations were calculated from Eq. (6) and are listed in Table 1. Comparison to a simpler model in which the inner cladding is approximated by solid silica indicates that the effect of the honeycomb is to increase

_{0}*S*by about 10–30%. The reason is that in a honeycomb offers a lower resistance to the pull exerted by the higher thermal expansion jacket than solid silica, thus the fiber length expansion is increased (larger

_{L}*S*). The effect of the honeycomb is thus small but not negligible. Table 1 also lists the values of the parameters used in our simulations for the other fiber layers.

_{L}The local deformation vector **u**(**r**) at the point **r** = [*r*, *θ*, *z*] is:

Only the diagonal components of the strain tensor ε are non-zero:

We use Hooke’s law to relate the stress tensor *σ* and strain tensor *ε* along with the effect of a temperature change Δ*T*:

where *s* is the fourth-order compliance tensor, *α* is the thermal expansion tensor, which also only has diagonal terms, and : denotes the tensor product.

It can be shown that the deformation field *u _{z}* does not vary with

*r*and that for a long fiber, it varies linearly with

*z*, i.e., it is of the form:

where *C* is a constant and the *z* origin is chosen in the middle of the fiber. Because *u _{z}*(

*z*) is continuous at each interface between layers,

*C*has the same value for all layers. Since the temperature is assumed uniform across the fiber, Eqs. (9–10) imply that

*ε*and

_{zz}*σ*are independent of

_{zz}*r*and only functions of

*z*. Furthermore,

*u*satisfies the admissibility condition:

_{r}whose solution is:

where *A* and *B* are constants specific to each layer. We solve for the coefficients *A*, *B*, and *C* by imposing the following boundary conditions and making use of Hooke’s law:

- continuity of
*u*(_{r}*r*) across all inner layer boundaries; - continuity of
*σ*(_{rr}*r*) across all inner layer boundaries; *σ*(_{rr}*r*) = 0 at*r*=*a*_{0}, where*a*_{0}is the fiber core radius;*σ*(_{rr}*r*) = 0 at*r*=*a*, where_{M}*a*is the outer radius of the fiber;_{M}- mechanical equilibrium on the fiber end faces, which imposes :

We use a matrix method to determine *A*, *B*, and *C*, and thus *u _{r}*(

*r*) and

*u*(

_{z}*z*). Equation 8 then yields the strains, including

*ε*=

_{zz}*S*Δ

_{L}*T*.

To illustrate the kinds of predictions we can make with this model, we plotted in Fig. 2 the radial deformation as a function of distance from the fiber center calculated for the Crystal Fibre PBF (parameters listed in Table 2). We observe that over the honeycomb structure (5 μm ≤ *r* ≤ 33.5 μm) and the silica outer cladding (33.5 μm < *r* ≤ 92.5 μm) the radial deformation remains small compared to the deformation of the acrylate jacket (92.5 μm < *r* ≤ 135 μm), which is consistent with the differences in the thermal expansion coefficient and stiffness of the materials. The low-thermal expansion and stiff silica experiences a much weaker deformation than the high-thermal expansion and soft acrylate. We also note that since the radial strain is the derivative of the radial deformation, the inner cladding honeycomb is under compressive strain, and it relaxes the strain over the structure by absorbing some of the deformation (the radial deformation decreases by a factor of 4 over the honeycomb structure) thanks to its very small transverse Young modulus.

Now that the strain distributions are known, computation of *S _{n}* is straightforward. In a first step, from the radial strain distribution we compute the change in the dimension of each layer across the fiber cross-section. In a second step, from the total strain distribution we compute the change in refractive indices of each layer due to the elasto-optic effect. Finally, in a third step we calculate the change in material indices induced by the temperature change (thermo-optic effect), which is independent of the strains and the easiest to evaluate. These three contributions (change in index profile, core radius, and materials’ indices) are then combined to obtain the refractive index profile of the fiber at

*T*=

*T*

_{0}+

*ΔT*. This new profile is then imported in the SPBF code [6], a finite difference PBF mode solver, to compute the effective index of the fundamental mode at this temperature. The code is also used to compute the mode effective index of the unperturbed fiber, i.e., at temperature

*T*

_{0}. Finally, these two values of the effective index are used in Eq. 3 to compute

*S*. This calculation assumes that all parameters change linearly with temperature, which is reasonable for small temperature excursions.

_{n}Figure 3 shows the dependence of *S _{L}*,

*S*, and

_{n}*S*on the core radius

*R*predicted by this model for a fiber with a cladding air-hole radius

*ρ*= 0.495Λ, an outer cladding radius of 92.5 μm, and an acrylate jacket of thickness 42.5 μm (parameters of the Crystal Fibre PBF). The signal wavelength was

*λ*= 0.5Λ, close to the middle of the fiber bandgap, with Λ = 3 μm. The values of

*S*,

*S*, and

_{L}*S*calculated for an SMF28 fiber (see parameters in Table 2) at

_{n}*λ*= 1.5 μm are also indicated for comparison. As expected,

*S*is almost independent of core radius and is the dominant term. The situation is reversed from a conventional SMF28 fiber, for which

_{L}*S*is significantly larger than

_{n}*S*(see Fig. 3). Note also that

_{L}*S*is sensibly the same for the air-core and the SMF28 fibers. The physical reason is that

_{L}*S*quantifies linear expansion of the fiber, which is similar in both fibers.

_{L}*S*is actually a little lower for the air-core fiber because of the increased relative area of silica in the outer cladding compared to the acrylate in the jacket for this particular fiber. Therefore the PBF has a lower overall thermal expansion than the SMF28 fiber. For the air-core fiber, the index term

_{L}*S*generally decreases slowly with increasing core radius (see Fig. 3), except for prominent local peaks in the ranges of 1.1Λ–1.25Λ and 1.45Λ–1.65Λ, where

_{n}*S*and

_{n}*S*increase by as much as a factor of two. These ranges coincide precisely with the regions when surface modes occur (highlighted in gray in Fig. 3).[7] The reason is that for the core radii that support surface modes, a significantly larger fraction of the fundamental mode energy is contained in the dielectric portions of the fiber, and the phase is more sensitive to temperature. This result points out yet another reason why surface modes should be avoided. Outside of these surface-mode regions, the total phase thermal constant

*S*=

*S*+

_{L}*S*varies weakly with core radius (see Fig. 3). The lowest

_{n}*S*value in the single-mode range (

*R*< ~1.1Λ) occurs for

*R*≈ 1.05Λ and is equal to ~1.68 ppm/°C, which is 4.9 times smaller than for an SMF28 fiber.

#### 2.2 Approximate weighted-average model

Since in an air-core fiber most of the contribution to *S* comes from the length term *S _{L}*, the more complex index term

*S*can be neglected and it is worth developing a simple model to evaluate

_{n}*S*(and thus

_{L}*S*). We can approximate

*S*to a good accuracy, while gaining some physical insight for the effects of the various parameters, by ignoring the radial terms and writing down the condition that the total force exerted on the fiber in the

_{L}*z*direction is zero. Using the notation in Fig. 1, this total force is:

where the subscripts *h*, *cl*, and *J* stand for honeycomb, outer cladding, and jacket, respectively (the corresponding term for the air core is zero and is thus absent from Eq. 14). Substituting Eq. (9) in Eq. (14) while neglecting the transverse terms, which are small because the fiber radius is small compared to the fiber length, we obtain the following approximate expression for *S _{L}*:

As can be seen from Table 1, the jacket expansion term *A _{J}E_{J}α_{J}* and the outer cladding expansion term

*A*are comparable in size and much larger than the honeycomb term A

_{cl}E_{cl}α_{cl}_{h}E

_{h}α

_{h}, which can be neglected. In the denominator, the main term is the restoring force

*A*due to the outer cladding, which is much larger than the force from the jacket or the honeycomb. Hence, Eq. (13) can be well approximated by:

_{cl}E_{cl}This simple expression shows that the best strategies to lower *S _{L}* are to make the area of the outer cladding

*A*as large as possible relative to the jacket area

_{cl}*A*, and to use a jacket material with a low thermal expansion. This approximate model turns out to be quite accurate, as will be illustrated in Section 6.

_{J}## 3. Experimental setup

The parameter *S* was measured for two PBF fibers, namely the AIR-10-1550 fiber manufactured by Crystal Fibre[8] and the HC-1550-02 fiber from Blaze Photonics (now Crystal Fibre).[9] SEM photographs of the fibers’ cross-sections are shown in Fig. 4. Measurements were carried out using the conventional Michelson fiber interferometer illustrated in Fig. 5(a). The signal source was a 1546-nm DFB laser with a linewidth of a few MHz. The air-core fiber was either spliced (Blaze Photonics fiber) or butt-coupled (Crystal Fiber fiber) to one of the ports of a 3-dB coupler (SMF28 fiber) to form the “sensing” arm of the interferometer. The far end of the PBF was similarly coupled to a fiber-pigtailed Faraday rotator mirror (FRM) to reflect the signal back through the fiber and thus eliminating polarization fluctuations in the return signal due to variations in the fiber birefringence. Most of the PBF was attached to an aluminum block placed on a heating plate, and the fiber/block assembly was covered with a styrofoam thermal shield (see Fig. 5(a)) to maintain the fiber temperature as uniform as possible and to reduce temperature fluctuations due to air currents in the room. The temperature just above the surface of the block was measured with a thermocouple.

The second (reference) arm of the interferometer consisted of a shorter length of SMF28 fiber splice to a second FRM. Together with the non-PBF portion of the sensing arm, this entire arm was placed in a second thermal shield (see Fig. 5(a)), mostly to reduce the amount of heating by the nearby heater of both the reference fiber and the non-PBF portion of the sensing arm. With this arrangement, when the heater was turned on the PBF was the only portion of the interferometer that was significantly heated.

To measure *S*, the temperature of the PBF was raised to around 70°C, then the heater was turned off and as the PBF temperature slowly dropped, both the output power of the interferometer and the fiber temperature were measured over time and recorded in a computer. During the measurement time window (typically tens of minutes), the phase in the PBF arm decreased and passed many times (50–200) through 2*π*, so that the power at the interferometer output exhibited many fringes, as illustrated in the typical experimental curves of Fig. 6. The phase thermal constant *S* was calculated from the measured number of fringes occurring in a given time interval using:

where *L* is the length of fiber under test and *ΔT* is the temperature change occurring during the measurement interval (see Fig. 6).

This approximation is justified because the temperature drop was slow enough that the PBF temperature was uniform at all times, yet fast enough that random phase variations in the rest of the interferometer were negligible compared to the phase variations in the PBF. To verify this last point, we measured the inherent temperature stability of the interferometer by disconnecting the PBF and reconnecting the fiber ends of the sensing arm with a short length of SMF28 fiber, as illustrated in Fig. 5(b). In a first stability test, we recorded the interferometer output while the entire interferometer temperature was at equilibrium with room. Over a period of ~30 minutes, the enclosure temperature was found to vary by ±1°C and the output power by about one fringe only. This test showed that the interferometer was more than stable enough to measure phase shifts of tens of fringes. In a second test, the PBF enclosure was heated to around 70°C, then the heater was turned off and the interferometer output was recorded as the heater slowly cooled down. This time we observed a larger number of fringes, which indicated that a little heat from the heater reached through the interferometer shield and induced a differential temperature change in the two arms. The output power varied by ~12 fringes while the enclosure temperature dropped ~18°C. Consequently, when measuring *S* with the setup of Fig. 5(a), residual heating of the non-PBF portion of the interferometer introduces an error of ~12 fringes. For this error to be small compared to the fringe count due to the change in the PBF temperature, this fringe count must be much larger than the error, e.g., 100 or more. This goal was met by using a long enough PBF. For the value of *S* ≈ 2 ppm/°C predicted for a PBF (see Fig. 3), Eq. 15 predicts that the length required to obtain 100 fringes of phase shift for a *ΔT* of 18°C is *L* ≈ 1 m. The length of PBF we used in our measurements was therefore of this order (about 2 meters, see Table 2).

## 4. Experimental results

As a point of comparison, we first measured the thermal constant of a conventional solid-core fiber by replacing the PBF in the experimental setup of Fig. 5(a) by a 210-cm length of SMF28 fiber. The measured value is *S* = 7.9 ppm/°C, in excellent agreement with the value of 8.2 ppm/°C predicted by our model using the parameter values of Tables 1 and 2. This value is the sum of *S _{L}* = 2.3 ppm/°C and

*S*= 5.9 ppm/°C, i.e., the index contribution is 2.6 times larger than the length expansion contribution. These values are summarized in Table 3. The close agreement between measured and calculated values gives credence to both our model and our interferometer calibration.

_{n}We then measured *S* for the two air-core PBFs. A typical experimental result is shown in Fig. 6. The value of *S* inferred for each fiber from such measurement and Eq. 15 is listed in Table 3, along with the calculated values of *S*, *S _{n}*, and

*S*. The

_{L}*S*values measured for the two PBFs are fairly similar, in the range of 1.5 to 2.2 ppm/°C. As predicted, the air-core fiber guidance mechanism results in a sizable decrease in the sensitivity of the phase delay on temperature. This reduction is as much as a mean factor of 5.26 (measured) or 5.79 (predicted) for the Crystal Fibre PBF. The corresponding figures for the Blaze Photonics fiber are 3.6 (measured) and 3.14 (predicted). Again, the theoretical and measured values agree well. The Crystal Fibre fiber exhibits a lower thermal expansion contribution than the Blaze Photonics fiber because it has a larger area of silica cladding relative to the jacket area (see Eq. 14). These reductions in

*S*result mostly from a decrease in

*S*by a factor of ~100, as well as a 15%–45% reduction in

_{n}*S*; as predicted by theory, in a PBF

_{L}*S*is determined overwhelmingly by

*S*, which depends only on the change in fiber length. The conclusion is that current air-core fibers are substantially less temperature sensitivity that conventional fibers, by a factor large enough (3.6–5.3) that it will translate into a significant stability improvement in fiber sensors and other phase-sensitive fiber systems.

_{L}## 5. Air-core fiber designs with further reduced thermal sensitivity

In this section we show that even smaller values of *S* can be obtained with improved PBF designs. Since in a PBF *S _{L}* is the main contribution to

*S*, to further reduce

*S*we must first try to reduce

*S*. This term arises from the thermal change in the fiber length, which is driven by both the thermal expansion coefficient and the stiffness of (1) the honeycomb cladding (silica and air), (2) the outer cladding (silica), and (3) the jacket (typically a polymer). Because polymers have a much higher thermal expansion coefficient than silica, as the temperature is increased the jacket expands more than the fiber, and thus it pulls on the fiber and increases its length more than if the fiber was unjacketed. The jacket is therefore generally the dominant contribution to

_{L}*S*. Consequently, a thinner jacket will result in a smaller

_{L}*S*, the lowest value being achieved for an unjacketed fiber. Furthermore, everything else being the same a softer jacket (lower Young modulus) will stretch the fiber less effectively and thus yield a lower

_{L}*S*. Finally, increasing the outer cladding thickness increases the overall stiffness of the fiber structure, thus reducing the expansion of the honeycomb and reducing

_{L}*S*.

_{L}These predictions were confirmed by simulating the Blaze Photonics fiber for various acrylate jacket thicknesses and air filling ratios (see Fig. 7). As expected, as the jacket thickness is reduced *S _{L}* decreases. In the limit of zero jacket thickness (bare fiber),

*S*reaches its lowest limit, set by the thermal expansion of the silica cladding. We also observe that for higher air filling ratios,

_{L}*S*is larger. The reason is that the honeycomb then contains less silica, the fiber has a lower overall stiffness, and the jacket expansion is less restrained by the glass structure, resulting in a larger

_{L}*S*value.

_{L}The effect of the jacket material stiffness can be seen in Fig. 8, where we graphed the calculated values of *S _{L}* (for the same PBF, with an air filling ratio of 90%) for a few standard jacket materials (metals, polymers, and amorphous carbon covered with polyimide[10]). To simulate actual fiber jackets, the jacket thickness was taken to be 5 or 50 μm for polyimide (as specified), 20 μm for metals, and 200 nm for amorphous carbon (covered with either 2.5 or 5 μm of polyimide). The reference jacket was 50 μm of acrylate. All metal jackets yield a larger

*S*than the reference acrylate jacket (2.57 ppm/°C). The explanation is that while metals have a lower thermal expansion than acrylate (by about one order of magnitude), their Young modulus is much larger than that of both acrylate (by 2–3 orders of magnitude) and silica (by a factor of up to 3). The silica structure is therefore pulled more effectively by the expanding metal coating than by the acrylate jacket, and

_{L}*S*is larger. Several jacket materials, however, perform better than acrylate. A thin (200 nm) amorphous carbon coating with a 2.5-μm polyimide jacket over it gives the lowest and best value,

_{L}*S*= 0.67 ppm/°C (74% reduction), followed by a 5-μm polyimide jacket (

_{L}*S*= 0.77 ppm/°C, 70% reduction). This is close to the theoretical limit for a silica fiber, which is set by the thermal expansion coefficient of silica and is equal to

_{L}*S*= 0.55 ppm/°C. The polyimide jacket performs best only because it is much thinner than an acrylate jacket. For equal thickness, acrylate actually performs better than polyimide. But because polyimide is a better water-vapor barrier than acrylate, a polyimide jacket only a few microns thick is sufficient to effectively protect the fiber against moisture, which is not true for acrylate. The conclusion is that acrylate unfortunately happens not to be the best choice of jacket material for thermal performance. By coating the PBF with the above carbon–polyimide jacket instead, an

_{L}*S*of only 0.67 ppm/°C, i.e., an

_{L}*S*as low as 0.72 ppm/°C, should be attainable, which is ~11 times lower than for a conventional fiber.

The effect of the silica outer cladding was also studied by simulating the same PBF for increasing cladding thicknesses, assuming a fixed 50-μm acrylate jacket and a 90% air-filling ratio. The result is plotted as the solid curve in Fig. 9. As expected, as the outer cladding thickness is increased *S _{L}* drops, because a thicker silica cladding better resists the length increase of the acrylate jacket. This effect is fairly substantial. For example, by doubling the outer cladding thickness from the 50-μm value of the Blaze Photonics fiber to 100 μm,

*S*is reduced by 55%. In the opposite limit of no outer cladding (zero thickness),

_{L}*S*jumps up to more than 20 ppm/°C. The high thermal expansion jacket is then pulling only on the silica honeycomb structure, which has a lower Young modulus due to the air holes and thus offers less resistance to stretching. Using a thick outer cladding is therefore an excellent way of reducing the thermal sensitivity of an air-core fiber. The downside is that the fiber is then stiffer and can therefore not be wound as tightly, which is a disadvantage in some applications.

_{L}The dashed curve in Fig. 9 was generated using our approximate model (see Section 3.2). This curve is in very good agreement with the exact result. Since again *S _{L}* accounts for more than 90% of the thermal constant

*S*, this very simple model is a reliable tool to predict the thermal constant of any fiber structure.

In summary, it is possible to reduce the thermal constant below the low value already demonstrated in existing air-core fibers by using (1) a jacket as thin as possible; (2) a soft jacket material; (3) a large outer cladding; and/or (4) a small air filling ratio (inasmuch as possible). Obvious jacket materials that satisfy (1) and (2) are polyimide and amorphous carbon covered by a thin layer of polyimide. With a 5-μm polyimide jacket, which is the best of these options, the Blaze Photonics fiber is predicted to have a thermal constant of *S* ≈ 0.82 ppm/°C, which is ~3.2 times smaller than in the current fiber.

We also computed the theoretical thermal phase sensitivity to temperature for a Bragg fiber with an core radius of 2 μm, surrounded by 40 air-silica Bragg reflectors with thicknesses of 0.48 μm (silica) and 0.72 μm (air), with an acrylate jacket thickness of 62.5 μm. This fiber exhibits a fundamental mode confined in its air core with a radius of 1.5 μm. We found *S _{L}* = 1.15 ppm/°C,

*S*= 0.30 ppm/°C, and

_{n}*S*= 1.45 ppm/°C (see Table 3). As expected again, because the fundamental mode in a Bragg fiber travels mostly in air this value of

*S*is much lower than for a conventional fiber.

*S*is comparable to the value for a PBF, and the main contribution is again the lengthening of the fiber.

## 6. Conclusions

The temperature sensitivity of the phase of the fundamental mode is a critical factor in the performance of interferometric fiber sensors such as the fiber optic gyroscope. This sensitivity is quantified by the parameter *S*, which is the relative change in phase per degree C change. This parameter is the sum of two contributions, a term due to fiber lengthening and a term due to thermal variation of the mode effective index. In an air-guiding PBF, because the fundamental mode is confined mostly in air, as opposed to silica in a conventional index-guiding fiber, we expect a lower value of the mode effective variation and therefore a reduced *S*, which should benefit many fiber applications.

To confirm and quantify this prediction, we first developed a thermo-mechanical model of the dependence of the fundamental-mode phase on temperature in air-core and conventional fibers, then we measured *S* using a Michelson interferometer in two commercial air-core fibers (from Crystal Fibre and Blaze Photonics) and a conventional fiber (Corning’s SMF28 fiber). For the SMF28 fiber, we measured *S* to be 8.2 ppm/°C, with the mode effective index variation being the dominant term. The corresponding values for the Crystal Fibre and Blaze Photonics fibers are 1.5 and 2.2 ppm/°C, respectively, which is substantially lower (factor of 3.6–5.3) than for the index-guiding fiber. These values are in good agreement (within 20%) with the values predicted by the model. The latter shows that the lower *S* of PBFs results from a drastic reduction (~100-fold) in their effective index variation, as expected at the outset. The residual thermal constant of an air-core fiber therefore arises almost entirely from the thermally induced lengthening of the fiber.

Finally, we show that with proper design of the fiber structure (thinner polymeric jacket and thicker outer cladding), S can be further reduced down to about 0.72 ppm/°C, or a factor of ~11 smaller than a conventional solid-core fiber. This greatly reduced temperature sensitivity is anticipated to have important practical benefits for many fiber sensors, especially the fiber optic gyroscope.

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