Graphene is a honeycomb-shaped sheet of carbon atoms arranged. It is one-atom thick (∼ 0.34 nm) and has attractive properties, such as good transparency,^[1,2] medium notable nonlinearity,^[3] broadband polarization,^[4,5] and strong conductivity.^[6] Its conductivity closely relates to the electromagnetic wave frequency, surrounding temperature and the Fermi level of the dopant, which makes graphene have great potential applications in nano integrated devices of communications and sensing.^[7–10] All optical fiber sensors have received great research interest and have been wildly applied to biological, chemical, and environmental industries, due to their unique advantages of high sensitivity, compact size, low cost, and immunity to electromagnetic interference, and so on.^[11,12] So far, there are some demonstrations of the combination of graphene and varieties of optical fiber sensing structures, such as titled optical fiber grating,^[13] etched fiber Bragg grating,^[14] and microfibers.^[15] By using various optical properties of graphene, some graphene coated devices are obtained. Because of the interactions between the excited backward propagating cladding modes of the tilted optical fiber grating and the grapheme film, some humidity and refractive index sensors are proposed.^[16–18] Optical biosensing and gas-sensing devices based on the etched fiber Bragg gratings coated with graphene with high sensing sensitivity, relative low attenuation, and large dynamic range are demonstrated.^[18,19] Microscalegraphene-coated microfibers or side polished fiber as saturable absorbers, light controller, wavelength converter, and sensors are studied.^[19–23] Compared with optical fiber Bragg gratings, in-fiber Mach–Zehnder and Michelson interferometer sensors have their own unique advantages such as high sensitivity, low cost, easy fabrication, etc. In 2016, the complex refractive index (CRI) of the graphene was demonstrated by detecting the phase and intensity variation of the Raman spectrum of a two-arm Mach–Zehnder interferometer, in which the sensing arm is attached to a graphene substrate.^[24] In 2016, a temperature induced refractive index (RI) influence of graphene-assisted microfiber temperature sensor from 30 °C to 80 °C was demonstrated by detecting the optical power of the Raman spectrum varying with temperature.^[7] However, microfibers are often used in these reports, which are drawn from the ∼ 1-μm-diameter and ∼ 2-mm-long single-mode optical fiber (SMF) with poor robustness sensing structure and complex fabrication process. In addition, the Raman spectrum of the microfiber is not at the conventional communication wavelength, which makes it an inconvenience to the application of conventional optical fiber active and passive devices in these devices, thereby increasing the cost of the systems.

In this paper, the effect of temperature on the refractive index of graphene based on a graphene coated in-fiber Mach–Zehnder interferometer (MZI) is demonstrated. In this study, grapheneis coated on the cladding that is partly etched by MTM (single mode-multimode-thined core-multimode-single mode) in-fiber MZI, through the direct deposition method. The temperature-induced effect of the refractive index of graphene is demonstrated through comparing the interference spectra between graphene coated MZI and uncoated MZI indifferent ambient temperatures in a range from 40 °C to 300 °C. It is found that the temperature can cause the CRI of the graphene to change. This temperature effect makes the real and imaginary parts of the CRI of the graphene change with the temperature. By measuring the variation of the interference spectrum of MZI, the relationships between the real and imaginary parts and temperature are obtained. This effect is also theoretically analyzed, which shows that the simulation and experimental results agree with each other. These results are also adoptable in the other CRIs of carbon nanomaterials such as carbon nanotubes and fullerenes. Meanwhile, this graphene-coated in-fiber MZI structure also presents special superiorities including easy fabrication, miniaturization, low cost, and robustness.

As shown in Fig. 1, when the light is injected into the MZI from the input SMF to MMF1, multiple modes will be excited and propagate within it. The light of these multiple modes is coupled from the MMF1 into the TCF, part of the multimodes light enters into the TCF cladding while the rest is coupled into the TCF core. In order to increase the interaction area between the cladding modes and the graphene coating, the cladding of the TCF is etched. Similarly, at the splice point between TCF and MMF2, part of cladding modes of the graphene coating TCF are again coupled into the MMF2, multiple modes of MMF2 are consequently coupled into the lead-out SMF, these cladding and core modes will interfere in the core of the lead-out SMF. This structure can be considered as an in-fiber Mach–Zehnder interferometer and its transmission spectrum can be expressed as^[8,16] 1 where I_core and I_clad are the light intensities of the core mode and the cladding modes in the core of the lead-out SMF, respectively; n_eff,co and n_eff,cl are the effective refractive indices of the core and cladding modes, respectively; L is geometry length of the TCF; λ is the wavelength of the propagating light.

From Eq. (1), when the interference transmission spectrum intensity reaches its minimum, the wavelength of the m-th order dip wavelength can be written as 2 The sensing section in this MZI structure is graphene coating TCF, and its sensing mechanism is analyzed in theory as follows: the electron-hole concentration of the graphene will increase with its surrounding temperature rising due to thermal excitation, resulting in the rising of the conductivity. The optical conductivity of graphene can be written as^[8,24] 3 where σ_intra and σ_inter are the intra-band electron-photon scattering and the direct inter-band electron transition conductivity of the graphene, respectively. These two conductivities can be evaluated from 4 5 where e is the electron charge, K_B = 1.38065 × 10⁻²³ J/K is the Boltzmann constant, T is the temperature, ħ = 1.0546 × 10⁻³⁴ J·s is the reduced Plank constant, ω is the radiation frequency, is the momentum relaxation time, with v_F ≈ 10⁸ being the Fermi velocity, m_u ≈ 10⁴ cm²/V·s the impurity-limited direct current mobility, and μ_c = 0.2403 × 10⁻¹⁹ the chemical potential.

Consequently, the complex conductivity of graphene is 6 where σ_r and σ_i are the real part and imaginary part of the conductivity of the graphene. They can be calculated from 7 8 From Eqs. (7) and (8), we can see that the real part and imaginary part of the conductivity of the graphene relate to temperature, the modeling results are shown in Figs. 6(a) and 6(b) with temperature T ranging from 0 to 300 °C. From Fig. 6, it can be found that the real part and imaginary part of the conductivity of graphene both nonlinearly increase with rising temperature.

Fig. 6.

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	Fig. 6. (color online) Plots of (a) σr and (b) σi versus temperature in a range from 0 °C to 300 °C.

With the relationship between the conductivity and the complex effective electrical permittivity, the complex electrical permittivity can be expressed as^[8,24] 9 where ε₀ = 8.85 × 10⁻¹² F/m is the vacuum dielectric constant, and d is the thickness of the graphene film on the TCF. The effective refractive index of the graphene can be written as 10 The complex refractive index of graphene also has its real part and imaginary part, which can be calculated from 11 12 where n_gr and n_gi are the real and imaginary part of RI of graphene.

Figure 7 shows the simulations of n_gr and n_gi changing with temperature in a range from 0 °C to 300 °C. From the simulation results, we can see that the real part of CRI nonlinearly decreases and the imaginary part of CRI nonlinearly increases with rising temperature.

Fig. 7.

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	Fig. 7. (color online) Plots of ngr and ngi versus temperature.

Let n_eff = n_gr + in_gi, then the transmitting wave in the graphene will be able to be written as 13 where E₀ is the initial amplitude of electric field amplitude, e_n is the unit vector in the direction of electric field vibration, r is the spatial coordinate position along the wave propagation direction, c is the speed of light in a vacuum, t is the time, and ϕ₀ is the initial phase of the light wave. In this equation, for transmitting wave, the exp[ω(−n_gi · r/c)] part is the attenuation coefficient, and the exp[iω(n_gi · r/c + t) + ϕ₀] part is the phase change of the interference spectrum.

Combining Eqs. (1) and (2), we can obtain conclusions that the intensity and the dip wavelength of the transmission interference spectrum of graphene-coating in-fiber MZI will be modulated by changing the complex RI of the graphene.

For the uncoated MZI, as shown in Fig. 4, the dip wavelength and intensity change with temperature, which can be explained as follows: when the surrounding temperature around the TCF increases, both the effective RI of the core and cladding modes increase. Due to the fact that the thermos-optic coefficient of the Ge-doped silica core is higher than that of the cladding consisting of fused silica. From Eq. (2), we can find that the effective RI difference between the core mode and the cladding modes become large with temperature increasing, which will make the dip wavelength of the interference spectrum of the uncoated MZI red-shifted. From Eq. (1), the relationship between the intensity of the uncoated MZI interference spectrum and the temperature can also be explained.

For the graphene coated MZI, the experimental results show that there are temperature characteristics obviously different from those for the uncoated MZI. From Figs. 5(a)–5(c), it can be found that the sensor interference dip wavelength presents a nonlinear red-shift with temperature increasing from 40 °C to 300 °C. The intensity of interference dip wavelength nonlinearly decreases with rising temperature. These can be explained as follows: the effective RI of cladding (n_eff,cl) of the graphene coated MZI in Eq. (2) can be seen as the effective RI of the graphene. From Eq. (13), it can be known that the real part (n_gr) of the graphene CRI relates to the phase of the interference spectrum of the graphene coated MZI, and from the simulation results of Fig. 7, the temperature increase can cause the real part (n_gr) to nonlinearly decrease. As a result of these temperature effects, the effective RI difference between core mode and cladding modes nonlinearly increase with rising temperature, as shown in Fig. 5(b). The relationship between interference spectrum intensity and temperature can be obtained from Eq. (1). For graphene coated MZI, the spectral intensity relates to the imaginary part (n_gi) of the graphene CRI as indicated in the exp[ω(−n_gir/c equation. From Eq. (1), we can find that the intensity will nonlinearly decrease with increasing temperature as shown in Fig. 5(c). We can obtain a very good quadratic fitting curve between dip wavelength and temperature with a 99.95% fitting degree. From this quadratic fitting cure function, the temperature-induced imaginary part for a transmitting wavelength range from 1531 nm to 1565 nm can be calculated. From the quadratic fitting curve functions of intensity and temperature and considering the imaginary part, the real part can also be calculated.

In conclusion, the temperature-induced CRI effect of graphene is demonstrated theoretically and experimentally through investigating the interference spectrum dip wavelength and intensity of a graphene coated in-fiber MZI changing with its surrounding temperature in a range from 40 to 300 °C. From our simulation results, it can be found that the complex conductivity of graphene can be expressed by a nonlinear relationship between the real and imaginary components of graphene conductivity and the temperature. On this basis, the relationship between the complex RI of the graphene and the temperature is also simulated, and the obtained results show that the real part of the graphene CRI nonlinearly increases and its imaginary part nonlinearly decreases with temperature increasing. In our experiments, a nearly 2-μm-thick graphene film is deposited on the surface of the cladding etched TCF of an in-fiber MZI. Its SEM images show that this film tightly adheres to the optical fiber. Experimental results show that the temperature-induced real part of the CRI of graphene relates to the attenuation coefficient, which causes the intensity to nonlinearly decrease with temperature rising, and the temperature-induced imaginary part of the CRI of graphene relates to the phase change, which causes the dip wavelength to nonlinearly change with temperature. This method can be used not only to detect the temperature-induced CRI of graphene, but also to design graphene-based optic devices for communication and sensing applications.