Consider a two-mode optical fiber of length z characterized by the wavelength-dependent OPD between x-polarized modes Δ 10 x ( z ; λ ). The spectral intensity resolved at the output of the two-mode optical fiber by a spectrometer is [18] (1) I x ( z ; λ ) = I 0 x ( λ ) { 1 + V A x ( λ ) V x ( z ; λ ) cos [ 2 π / λ ] Δ 10 x ( z ; λ ) ] } ,where I 0 x ( λ ) is the reference (unmodulated) spectral intensity, V A x ( λ ) is an aperture visibility term and V^x(z; λ) is the visibility term given by (2) V x ( z ; λ ) = exp { − ( π 2 / 2 ) [ Δ 10 g x ( z ; λ ) Δ λ R / λ 2 ] 2 } ,where Δλ_R is the width of the spectrometer response function and Δ 10 g x ( z ; λ ) is the intermodal group OPD between the modes. The period of the spectral modulation Λ^x(λ) for the x-polarized modes is given by (3) Λ x ( λ ) = λ 2 | Δ 10 g x ( z ; λ ) | .

The equalization wavelength λ 0 x is resolved in the recorded spectrum for which the period is infinite, or equivalently the relation Δ 10 g x ( z ; λ 0 x ) = 0 is fulfilled.

Similarly, consider a two-mode optical fiber characterized by the wavelength-dependent OPD between y-polarized modes Δ 10 y ( z ; λ ). The spectral intensity resolved at the output of the two-mode optical fiber by a spectrometer is: (4) I y ( z ; λ ) = I y ( 0 ) ( λ ) { 1 + V A y ( λ ) V y ( z ; λ ) cos [ ( 2 π / λ ) Δ 10 y ( z ; λ ) ] } .

The period of the spectral modulation Λ^y(λ) for the y-polarized modes is given by (5) Λ y ( λ ) = λ 2 | Δ 10 g y ( z ; λ ) | ,where Δ 10 g y ( z ; λ ) is the intermodal group OPD between the modes. The equalization wavelength λ 0 y is resolved in the recorded spectrum for which the relation Δ 10 g y ( z ; λ 0 y ) = 0 is fulfilled.

Consider a linearly polarized optical field, propagating in the positive z direction along the axis of a lossless optical fiber (see Figure 1), in which only the fundamental mode in both x and y polarizations is excited. If the wavelength-dependent phase effective (refractive) indices of the fundamental mode are denoted as n_x(λ) and n_y(λ), we can define the phase modal birefringence (6) B ( λ ) = n x ( λ ) − n y ( λ ) .

Furthermore, using the relation for the group effective (refractive) index (7) N ( λ ) = n ( λ ) − λ d n ( λ ) d λ = − λ 2 d [ n ( λ ) / λ ] d λ ,we can express the group modal birefringence G(λ) (8) G ( λ ) = N x ( λ ) − N y ( λ ) = − λ 2 d [ B ( λ ) / λ ] d λ .

The spectral intensity at the output of the configuration shown in Figure 1 with a polarizer and an analyzer oriented 45° with respect to the fiber eigenaxes is [6] (9) I ( z ; λ ) = I 0 ( λ ) { 1 + V ( z ; λ ) cos [ ( 2 π / λ ) B ( λ ) z ] } ,where I₀(λ) is the reference spectral intensity and the visibility term V (z; λ) is given by (10) V ( z ; λ ) = exp { − ( π 2 / 2 ) [ G ( λ ) z Δ λ R / λ 2 ] 2 } .

The period of the spectral modulation Λ(λ) is given by (11) Λ ( λ ) = λ 2 | G ( λ ) z | .

If the resolving power of the spectrometer is insufficient to resolve spectral interference fringes, the fiber in tandem with a birefringent crystal of the phase B_c(λ) and group G_c(λ) birefringences can be used [6]. The spectral intensity at the output of the tandem configuration shown in Figure 1 with a polarizer and an analyzer oriented 45° with respect to the fiber eigenaxes is [6]: (12) I ( z ; λ ) = I 0 ( λ ) { 1 + V ( z ; λ ) cos { ( 2 π / λ ) [ B c ( λ ) d − B ( λ ) z ] } } ,where the visibility term V (z; λ) is given by (13) V ( z ; λ ) = exp { − ( π 2 / 2 ) { [ G c ( λ ) d − G ( λ ) z ] Δ λ R / λ 2 } 2 } .

The period of the spectral modulation Λ(λ) is given by (14) Λ ( λ ) = λ 2 | G c ( λ ) d − G ( λ ) z | ,

The equalization wavelength λ₀ is resolved in the recorded spectrum for which the relation G_c(λ₀)d = G(λ₀)z is fulfilled.

Using the experimental setup shown in Figure 1, we measured the spectral intermodal sensitivities K 10 ∊ x ( λ ) and K 10 ∊ y ( λ ) for x and y polarizations of the modes of the investigated fiber to strain. The parameters are defined by the following relations (15) K 10 ∊ x ( λ ) = 1 L d [ Δ ϕ 10 ∊ x ( λ ) ] d ∊ ,and (16) K 10 ∊ y ( λ ) = 1 L d [ Δ ϕ 10 ∊ y ( λ ) ] d ∊ ,and Equation (15) represents an increase in the phase shift between the two modes in x polarization induced by the unit change of strain ∊ acting on unit fiber length. To determine the spectral intermodal sensitivity K 10 ∊ x ( λ ) for x polarization of the modes of the investigated fiber to strain, we recorded a sequence of spectral interferograms for increasing strain ∊ with a step small enough to assure unambiguity in retrieving the strain-induced phase changes Δ [ Δ ϕ 10 ∊ x ( λ ) ]. To measure the parameter, the fiber of length L = 0.895 m was attached with epoxy glue to two translation stages and elongated up to 800 μm (stretched up to 894 μstrain). The measurements were repeated several times for increasing and decreasing strain with no hysteresis observed.

Figure 3(a) shows two examples of the recorded spectra obtained for x polarization of the modes and corresponding to two elongations ΔL₁ = 100 μm and ΔL₂ = 300 μm of the fiber when the overall length of the fiber, including the leading part with length z₁ = 0.565 m, the sensing part with length L = 0.895 m and the outer part with length z₂ = 0.835 m, was z = 2.295 m. It is clearly seen from the figure that the intermodal interference shows up as the spectral modulation with the wavelength-dependent period and the equalization wavelength λ 0 x = 565.89 nm. The spectral interference fringes for the two elongations are with the same equalization wavelength but with different phases. Using the procedure presented in a previous paper [19], we retrieved from the two spectral interferograms the phase functions Δ ϕ 10 ∊ x ( λ ) that are shown in Figure 4(a). These phase functions are wavelength dependent with a minimum at the equalization wavelength λ 0 x. From the two phase functions we retrieved the strain-induced phase change Δ [ Δ ϕ 10 ∊ x ( λ ) ] and the spectral intermodal strain sensitivity K 10 ∊ x ( λ ) defined by Equation (15). Figure 5(a) shows by the blue curve the absolute value of K 10 ∊ x ( λ ) as a function of wavelength. Figure 5(a) also shows by the red line the approximate linear dependence obtained from eight phase differences retrieved from eight recorded spectral interferograms. It is clearly seen from the figure that the absolute value of the spectral intermodal strain sensitivity K 10 ∊ x ( λ ) decreases with a wavelength (in a range of 530 to 600 nm) from a value of 4.2 rad· m⁻¹· m∊⁻¹ to a value of 1.7 rad· m⁻¹· m∊⁻¹.

Similarly, Figure 3(b) shows two examples of the recorded spectra obtained for y polarization of the modes and corresponding to two elongations ΔL₁ = 100 μm and ΔL₂ = 300 μm of the fiber. It is clearly seen from the figure that the intermodal interference shows up as the spectral modulation with the wavelength-dependent period and the equalization wavelength λ 0 x = 551.51 nm, which is shifted to the shorter wavelengths in comparison with that for x-polarized modes. The spectral interference fringes for the two elongations are with the same equalization wavelength but with different phases. We retrieved from the two spectral interferograms the phase functions that are shown in Figure 4(b). Figure 5(b) then shows by the blue curve the absolute value of the spectral intermodal strain sensitivity K 10 ∊ y ( λ ) as a function of wavelength. From the phase functions retrieved from all the recorded spectral interferograms the spectral intermodal sensitivities K 10 ∊ y ( λ ) were obtained. Figure 5(b) shows by the red line the approximate linear dependence obtained from eight retrieved phase differences. It is clearly seen from the figure that the absolute value of the spectral intermodal strain sensitivity K 10 ∊ y ( λ ) decreases with a wavelength and that it is approximately three time higher and with a greater slope than the spectral intermodal strain sensitivity K 10 ∊ x ( λ ).

It is clear from the experimental results that an implementation of a new sensor configuration is possible. In the experimental setup shown in Figure 1 the spectral intensity can be measured for different elongations and from the corresponding recorded spectra the phases can be retrieved. As an example, the retrieved phase can be with the elongation sensitivities similar to those shown in Figure 4(a,b).

Using the experimental setup shown in Figure 1 and including delay line DL, we measured the spectral polarimetric sensitivity K_∊(λ) of the fundamental mode of the investigated fiber to strain. The parameter is defined by the following relation (17) K ∊ ( λ ) = 1 L d [ ϕ x ( λ ) − ϕ y ( λ ) ] d ∊ ,and represents an increase in the phase shift between the two polarizations of the fundamental mode, i.e., between the polarization modes, induced by the unit change of strain ∊ acting on unit fiber length [10].

Figure 6(a) shows two examples of the recorded spectra obtained for a suitable chosen delay line and corresponding to two elongations ΔL₁ = 100 μm and ΔL₂ = 300 μm of the fiber when the overall length of the fiber, including also the sensing part with length L = 0.895 m, was z = 2.295 m. It is clearly seen from the figure that the interference of polarization modes in the tandem with the delay line shows up as the spectral modulation with the wavelength-dependent period and the equalization wavelength λ₀ = 581.61 nm. The spectral interference fringes for the two elongations are with the same equalization wavelength but with different phases. We retrieved from the two spectral interferograms the phase functions that are shown in Figure 7(a). These phase functions are wavelength dependent with a minimum at the equalization wavelength λ₀. From the phase functions retrieved from eight recorded spectral interferograms the spectral polarimetric sensitivity to strain K_∊(λ) for the fundamental mode was obtained. Figure 8(a) shows by the blue curve the absolute value of K_∊(λ) as a function of wavelength. Figure 8(a) also shows by the red line the approximate quadratic dependence obtained for eight phase differences retrieved from eight recorded spectral interferograms. It is clearly seen from the figure that the absolute value of the spectral polarimetric sensitivity to strain K_∊(λ) decreases with a wavelength (in a range of 500 to 700 nm) from a value of 21.8 rad· m⁻¹· m∊⁻¹ to a value of 16.0 rad· m⁻¹· m∊⁻¹. This sensitivity is approximately three times of that for a birefringent holey fiber [10].

We also measured the spectral polarimetric sensitivity to strain K_∊(λ) for the fundamental mode with a different delay line when no equalization wavelength λ₀ is resolvable in the recorded spectrum. In this case the spectral fringes of slightly wavelength-dependent period are resolved in the recorded spectrum and the spectral polarimetric sensitivity to strain K_∊(λ) is determined from the change of spectral phase with the elongation of the fiber. As an example, Figure 6(b) shows two examples of the recorded spectra corresponding to two elongations ΔL₁ = 100 μm and ΔL₂ = 300 μm of the fiber. It is clearly seen from the figure that the interference of polarization modes in the tandem with the delay line shows up as the spectral modulation with the wavelength-dependent period and with no equalization wavelength. The spectral interference fringes for the two elongations are with different phases which were retrieved from the two spectral interferograms using a procedure based on the application of a windowed Fourier transform [20]. The corresponding phase functions are shown in Figure 7(b). Figure 8(b) shows by the blue curve the absolute value of the spectral polarimetric sensitivity to strain K_∊(λ) for the fundamental mode as a function of wavelength together with the approximate quadratic dependence (red line) obtained for eight phase differences retrieved from eight recorded spectral interferograms. This approach gives the spectral polarimetric sensitivity to strain K_∊(λ) with errors (artefacts) smaller in comparison with those obtained by the first approach.