Spontaneous Raman Scattering is an inelastic scattering phenomenon, resulting from the interaction of light with the vibrational or rotational modes of molecules in the transmitting medium [5]. In the process, energy is transferred between the energy levels of the molecules constituting a material and the incident light [6]. This is more clearly depicted in Figure 2, which shows the energy diagram in Raman interactions where light with angular frequency of ω passes through a material where molecules have an initial energy of E_1.

As energy is transferred from the light to the vibrational modes of the material, the molecules go to transitional virtual states, which are not stable energy states. When the molecules transit to another state E₂, the differential energy is absorbed by the material and hence the scattered Stokes wave with angular frequency ω_s has energy less than that of the incident light by an amount given by [6]:

where ℏ=h/2π, and h = 6.62607004 ×10⁻³⁴ m² kg/s is Planck's constant. The equivalent interactions for the Raman anti-Stokes case is shown in Figure 3. In this case, first, upon an incident light with frequency of ω, the vibrational/rotational states of the material are transferred from E₁ to a transitional state, and then back to the ground state_. The energy difference between the two levels, which will be transferred to the incident light, results in anti-Stokes light with angular frequency of ω_AS, with a difference in energy given by [6]:

In the special case of the Raman scattering in a glass fiber, the Raman signal is located in a very wide band centered at ~13 THz from the input light, which corresponds to a shift of ~100 nm at 1,550 nm wavelength.

Brillouin scattering occurs due to the interaction of light with the sound waves or the acoustic phonons of a material. The SpBS phenomenon can be understood by considering the wave vector diagram shown in Figure 4A [7], which depicts the interaction between an incident and scattered light which respectively have wave vectors k and k′. The incident light interacts with the sound wave in the material which is represented by a vector q. As shown in Figure 4B, the magnitude of the sound wave vector for a generic scattering direction of ϕ will be given by [7]:

Assuming the frequency of the sound wave is much smaller than that of light [7], which will be justified later, the energy conservation allows the approximation|k|≅|k′|, and considering the trigonometric identity 1 − cosϕ = 2sin²(ϕ/2), we obtain:

To obtain the frequency of the sound wave, it is possible to employ the dispersion relation for the sound wave which connects its velocity and angular frequency Ω [8]:

where n is the phase refractive index. It can be seen from (5) that the Stokes shift has a sinusoidal dependence on the scattering wave vector angle and hence, there is no scattering in the forward direction, where ϕ = 0, while being non-zero for all other directions. Since light propagates in the longitudinal direction in the fiber, this in turn means that Brillouin scattering in an optical fiber is essentially Brillouin backscattering. The maximum value happens for ϕ = π, for which the frequency will have Ωs-max=2nυcω. For typical values of the parameters in a singlemode silica fiber, the speed of sound υ = 5,960 m/s, group index n = 1.5, λ = 1,550 nm, the maximum frequency shift due to Brillouin scattering can be calculated as [9]:

The anti-Stokes counterpart of the aforementioned process has a vector schematically described in Figure 5A. In this case, as energy is transferred from the soundwave to the incident light, the scattered light has a wave vector component which is the summation of the wave vectors of the incident light and acoustic wave. As per the dispersion relation, the frequency of the scattered light will be the sum of the two waves.

As shown in the schematic in Figure 5B, the scattered wave vector will be given by k' = k + q and the frequency will be ω′ = ω + Ω, where ω and Ω are the angular frequencies of the two waves related to the wave vectors via each other's dispersion relations. The magnitude of q will now be given by [7]:

Using the previous property |k| ≅ |k′|and inserting this in (7), the anti-Stokes frequency is also given by:

Hence, for singlemode silica fibers, the frequency shift due to the anti-stokes light is also the same as Ω ~11.536 GHz at 1,550 nm. Note that, like any other medium, the soundwave propagating in an optical fiber inevitably undergoes damping. This results in the scattering in any direction to have a certain bandwidth. The bandwidth of the Brillouin scattering can be given by [7]:

In (9), Γ′ represents the damping parameter of the acoustic wave, which gives a measure of the decay of the oscillations in a system. Note that, the Stokes and anti-Stokes components of Brillouin scattering signal have spectra which are symmetric and have similar bandwidth. Since they also have equal shifts in the two directions as shown in (5) and (8), the two components are located symmetrically at the two sides of the spectrum of the incident light, which is also depicted in Figure 1.

Stimulated Brillouin scattering takes place when the intensity of the incident light is high enough to result in a change in the density of the material through a mechanism known as electrostriction, whereby the material becomes denser in places where the gradient of the optical intensity is high. This mechanism is exploited in optical amplification as well as optical fiber sensing. When light with time varying electric field E passes through a medium, the change in the material density ρ is proportional to the dot product of the field and is given by [7]:

Where C = 1/K, is the compressibility with K being the bulk modulus, and γ_e the electrostrictive constant. To obtain a quantitative expression for the wave interactions in SBS, we consider the induced (i.e., stimulated) scattering from a seed Stokes signal via electrostriction, assuming a counter-propagating signal and Stokes waves in a generic medium such as an optical fiber, as shown in Figure 6.

For the general case in which the wave vector for the incident light is k and that of the counter-propagating Stokes wave is k_s, since the sound wave is enhanced by the interaction between the Stokes and incident light waves, its wave vector will be:

where vector sum in the calculation of the magnitude is made considering that the two waves are counter-propagating in the medium. Considering the angular frequency of an acoustic wave having velocity υ, Ω_B = |q_B|υ and inserting this in equation (11), while using dispersion relations for the signal |k| = nω/c and corresponding expression for the Stokes wave ω = ω_s + Ω, the acoustic wave will have angular frequency given by [8, 9]:

Equation (12) can be further expanded and solved, using the aforementioned dispersion relations to obtain an expression for the Brillouin frequency shift:

Note that for light waves typically used in optical fibers, the velocity of the acoustic wave is significantly small with respect to c/n. Hence, from the expression in (13), a good approximation to the Brillouin frequency shift can be ΩB=2υc/nωs.

The diagram in Figure 7 shows transfer of energy to and from the incident light and the two symmetric components. If the component at a higher frequency has an intensity of I_AS and the incident signal has an intensity of I_si, it can be shown that it will transfer energy to the pump along the longitudinal direction z with its intensity given by [7]:

In (14) g is the Brillouin gain:

where ΓB=1τp=q2Γ′ is the Brillion line width, with τ_p and Γ′ being the acoustic phonon life time and the damping factor, while q is the velocity of the acoustic wave and g_o is a material dependent constant.

Distributed measurement of temperature using SpRS is made possible due to the fact that the intensity of the anti-Stokes scattering optical fibers is dependent on the ambient temperature in the vicinity of the fiber. Since the Stokes component is only slightly dependent on the local temperature T at each spatial point z, the ratio between the two is given by [10]:

where Ω_R is the Raman frequency shift and ΔE = −ℏΩ_R is the energy difference between the incident and scattered light. The dependence of the ratio between the Stokes and anti-Stokes components on temperature comes from the population levels of the excited vibrational modes of the glass in thermodynamic equilibrium. The anti-Stokes process requires a pre-existing optical phonon, which is more likely to occur at higher temperatures. Specifically, the phonons in the different vibrational energy states follow Boltzman's distribution with temperature and the ratio of the anti-Stokes to Stokes intensities scales exponentially with temperature. The dependence on temperature is more for the anti-Stokes component compared to the Stokes one since the relative sensitivities ε=1IdIdT of the two are, respectively, 0.83% K^−f and 0.096% K^−a for typical silica-based glass at room temperature [11].

The use of SpRS for measurement of distributed temperature in time domain is known as Raman Distributed Temperature Sensing (RDTS), and its basic schematic is shown in Figure 8. Light from a pulsed laser is sent into the sensing fiber through a circulator and Raman filter with output at the Stokes and anti-Stokes lights is used to separate the two components of the backscattered light, which are detected using Avalanche Photodiodes (APDs), amplified using the Trans-impendence Amplifiers (TIAs) and acquired for further processing.

To account for the wavelength-dependent factor, the ratio in (20) is made at a reference temperature T_ref, and a reference ratio is made to yield:

Subsequently, (17) is rearranged to make distributed temperature measurement at each point along the fiber:

Measurement based on spontaneous Brillouin scattering hinges on the dependence of the Brillouin Gain Spectrum (BGS) on presence of heating or expansion of the fiber. Specifically, the frequency offset between the peak of the BGS and that of the spectrum of the incident light, also known as the Brillouin Frequency Shift (BFS), is linearly dependent on the strain or temperature at each point of the fiber, as shown in Figure 9. Hence, distributed measurement with this technique boils down to determining the distributed BFS along the fiber.

For a change in longitudinal strain Δε and that of temperature ΔT, the BFS has the form:

where the parameters C_ε and C_T are the strain and temperature coefficients of the specific fiber. They have typical values of ~0.05MHz/με and ~1MHz/°C for standard singlemode silica fibers at 1,550 nm.

A simple schematic for measurement using the BOTDR technique is shown in Figure 10. As shown, the CW signal from a narrowband laser is gated into optical pulses using an intensity modulator. The pulses are then sent into the Fiber Under Test (FUT) through the three-port circulator. The backscattering from the fiber at the return port of the circulator at the Stokes frequency is then allowed to beat with a local oscillator through a 3 dB coupler and detected using the balanced photodiode. The resulting beating is mixed with an RF local oscillator whose frequency is scanned across the possible range of the BGS in subsequent measurements. The LPF is used to remove high frequency components. The schematic shown in Figure 11 shows the scanning of the frequency of the mixer to reconstruct the plot of the BGS.

The basic mechanism of sensing using the BOTDA involves pump and probe signals which are applied at the two ends of the sensing fiber. As these signals counter-propagate and interact through electrostriction, energy is transferred from the pump to the probe via SBS. As shown in a basic schematic given in Figure 12, first a CW light from a single narrowband laser is allowed to pass through a coupler, which feeds the input light in two paths: an upper one constituting the path for the pulsed pump and the lower one for the CW probe.

The pulsed light is generated using a Mach-Zehnder Modulator (MZM) which acts as an intensity modulator and forms the pump which is fed to the sensing fiber through an optical circulator. The other part of the CW light is modulated with a sinusoidal RF signal at the Brillouin shift and will act as the CW probe, which is sent into the other end of the fiber. After proper bandpass filtering of the SBS signal at the Stokes frequency, the resulting trace is detected using a photodiode followed by a digital acquisition and processing block. The frequency of the CW probe, which has a nominal frequency with offset at the BGS shift of a singlemode fiber at around 11 GHz, is swept with respect to the center frequency of the pump.

The measurement mechanism involved in BOTDA is depicted in Figure 13, where the BGS of the fiber is determined for the whole fiber and the resulting plot becomes a three-dimensional plot of intensity vs. frequency for each spatial location along the fiber with a resolution determined by the pulse width. The distributed BFS is then calculated and exhibits an offset for any section which is either strained or heated.

Note that RDTS allows distributed temperature measurements without any cross-sensitivity with other parameters such as strain, while Brillouin-based sensors suffer from cross-sensitivity of temperature and strain which require other techniques to resolve. These include hybrid Raman/BOTDA schemes or hybrid Raman/BOTDR configurations. In addition, while simple RDTS involves an intensity-based measurement in which methods to account for non-local intensity changes are needed, Brillouin-based measurements are solely based on wavelength shift. As can be seen from Figures 10, 12, a key difference between BOTDA and BOTDR schemes is that the former requires both ends of the fiber as opposed to the latter in which one end suffices. It is worth noting that some RDTS schemes which address time-varying wavelength-dependent losses involve loop configurations requiring both ends of the fiber [12, 13]. Thanks to advanced schemes which include differential pulse pair, BOTDA is able to offer more spatially resolved temperature measurements compared to RDTS.

In conventional distributed sensors, only a single pulse is sent into the sensing fiber. However, in BOTDA or RDTS measurements, shorter pulses which enable spatially resolved measurements have less energy and vice versa. The peak power of the pulse can be increased for a higher energy but there is a threshold of power levels before the onset of other detrimental non-linear effects in long distance sensing. This entails a tradeoff between spatial resolution and sensing distance which needs resolving using advanced techniques. Detrimental non-linear effects include modulation instability and forward stimulated Raman scattering. In addition, there is also pump depletion, which ensues due to gradual power transfer from the higher frequency wave to the lower one along the fiber resulting in measurement errors due to a reduction in the pump power (in a gain configuration) at long distances [14].

One of the methods to resolve the resolution-distance tradeoff is the use of optical pulse coding. In this technique, instead of sending a single pulse per measurement round, a given pattern of pulses is sent within a single round-trip-time (RTT) as shown in the schematic given in Figure 14, where τ is the bit duration given by RTT/N, N being the length of the pattern, also known as a codeword. The resulting response will be a superposition of the delayed responses. The pulse patterns could be linearly combinational codes (e.g., Simplex codes) [15], or complementary correlation ones such as Golay codes. When the fiber is probed with a simplex codeword, the resulting response is equivalent to a matrix multiplication involving a simplex Matrix S_M representing the presence and absence of a pulse by either a 1 or a 0, respectively. If η_i(t) for i = [1, 2] represents the response of the signal for a given combination of the pulse, and y_i(t) = y(t-iτ) is the delayed response with e_i(t) being the error associated with each measurement, the matrix multiplication representing such a response can be written as:

Subsequently, the equivalent of the single pulse can be obtained from this sum by using an inverse matrix multiplication operation. The same can be done for a generic case of a codeword having a simplex matrix S_M of length M and it can be shown that for a simplex coded word of size M, the gain in SNR when using pulse coding when compared to a single pulse, also known as the coding gain, is given by [16]:

In complementary Golay correlation coding, first conceived by Naseem et al. [17] and Golay [18], each sequence in the set is such that the number of pairs of like elements with any separation in one sequence is the same as the number of pairs of unlike elements in another. Formally, a complementary correlation Golay code of sequence A and B with respective elements a_i and b_i, each of length L, must ensure that the respective autocorrelation functions c_i and d_i, given by [18]:

are such that:

where the expression in [19] is also commonly denoted by a delta function. Golay complementary codes are bipolar and, for use in optical systems which are unipolar, they must be represented in an equivalent form. This is typically done by defining the elements in each sequence corresponding to “1”s and “−1”s as A₁ = (1+A)/2, A₂ = (1−A)/2, and B₁ = (1+B)/2, B₂ = (1−B)/2 [17]. Note that the coding gain with respect to a single pulse when using unipolar Golay sequence is given by G=L/4. Among others, Golay coding has been used to suppress the impact of pump depletion in BOTDA with a coding gain G=L, with a more enhanced scheme based on three-tone bipolar Golay code offering increased robustness against this non-linear effect [20].

Hence, the use of optical pulse coding results in an SNR enhancement in the measurement with a given pulse width at low power levels, and hence enables more accurate measurements at long distances with meter-scale spatial resolution. A number of pulse coding schemes have been demonstrated in advanced BOTDA configurations [21, 22] and a representative scheme is shown in Figure 15 [21]. First, light from a narrowband Distributed Feedback (DFB) laser is coupled into two branches using a 30:70 coupler which results in the probe in the upper arm and the pump in the lower one. In the probe branch, the Mach-Zhender Modulator (MZM) is modulated with a sinusoidal signal whose frequency is scanned in steps of a few MHz. After amplification with an Erbium-Doped Fiber Amplifier (EDFA) and using a Variable Optical Attenuator (VOA) to adjust the power, a polarization scrambler is used to depolarize the light, and an isolator blocks light coming from the probe side.

On the lower branch of the pump, the light is first amplified and fed to an MZM which is used to generate the pulses. The high-peak power pulses are then sent through the circulator into a FUT comprised of 50-km standard single-mode fiber at the end of which a 1-m segment is heated. At the receiver, the backscattered signal is filtered using a combination of an FBG and circulator and the resulting traces are detected using a simple pin photoreceiver which feeds the data acquisition system composed of an oscilloscope. Then, subsequent post-processing is applied to reconstruct the BGS via a curve fitting algorithm for each point along the fiber. The technique is used to experimentally demonstrate distributed measurement at a spatial resolution of 1 m, with a temperature/strain accuracy of 2.2°C/44 με at 50 km.

Note that differences in the power levels of consecutive pulses can lower the accuracy of the measurement in coding at long distances [23]. The linear superposition of the contribution of the backscattering from long pulses could also be lost as successive pulses encounter a probe wave front which has been amplified by previous ones, resulting in the sensitivity of the pump depletion to the amplification of the wave font by previous pulses in the sequence [24]. In addition, use of long coded sequences has been demonstrated to lower the threshold of amplified spontaneous Brillouin scattering in the fiber. Techniques such as linearization using logarithmic processing of the probe response [25] allow maintaining superposition with coding while probe dithering has been demonstrated to address the limitations of pump depletion and lowered amplified spontaneous scattering threshold due to coding [24].

The application of optical pulse coding has also been extended to RDTS configurations. One such demonstration includes the use of low-repetition rate cyclic coding [19] and the experimental setup is depicted in Figure 16. As shown, first light from a pulsed narrowband laser is modulated using an Acousto-optic Modulator (AOM) driven with a pulse pattern generator before being sent into the FUT composed of a 26-km SMF using a Raman filter, which also separates the Stoke and anti-Stokes components of the Raman backscattering signal. The signal in each branch is then detected using an avalanche photodiode (APD), and acquired using FPGA-controlled analog to digital converter. This is followed by post-processing to extract the temperature from the two components. A 71-bit cyclic simplex coding is shown to result in a dynamic range enhancement of 6 dB, enabling fast measurement of temperature (within 30 s, as opposed to few minutes in other systems) with a resolution of 3°C at 26 km, with a spatial resolution of 1 m.

Note that one of the critical issues hampering the long-term stability of RDTS systems is the variation of Wavelength-Dependent Losses (WDL) with time due to various environmental factors. One way to address this issue is by using a loop configuration in which the geometric mean of the ratio of the backscattering intensities in the forward and backward directions is used to mitigate detrimental effects of WDL in reducing the accuracy of temperature measurements [19]. A method which uses two light sources at different emission wavelength is also demonstrated, whereby the anti-Stokes backscattering of one source spectrally overlaps with the incident light from the other and cancels out non-identical attenuations caused by wavelength differences between Stokes and anti-Stokes components [26]. An alternative scheme addresses WDL without the need for additional source employing a mirror at the end of the fiber which generates an anti-Stokes wave canceling out the effect of local attenuation [27]. There is also a more robust method to correct WDL via the measurement of anti-Stokes component alone in which the data processing is simplified by avoiding correction of the differential group velocity between the two Raman components [19]. There are also investigations of other advanced standalone BOTDA schemes for fast measurements using slope-assisted BOTDA [28], adaptive filtering [29], high spatial resolution using a Differential Pulse-Pair (DPP-BOTDA) [30], and dynamic Brillouin gratings (DBG), which have applications in birefringence measurement and all-optical processing [31].

Note that BOTDA suffers from cross-sensitivity between temperature and strain as shown in equation (19). Various configurations have so far been proposed and demonstrated to discriminate between of the effects of the change in BFS due to strain and temperature. Among others, a hybrid sensing scheme combining BOTDA in a loss configuration and SpRS requiring two separate lasers for the two scattering phenomena has been proposed [32], but it does not allow concurrent measurements as the source has to be used alternatively. A method for complete discrimination of strain and temperature based on a polarization maintaining fiber [33], and distributed measurement of Brillouin frequency shift and birefringence at short distances has been demonstrated [34] as well as one for simultaneous measurement of the Rayleigh and SpBS components [35]. Another investigation involved the study of fibers with multiple acoustic modes for simultaneous strain and temperature measurements [36]. A hybrid scheme involving the combination of the SpRS and SpBS has also been proposed but is limited by the SpBS intensity having low strain dependence in long distance sensing [37]. A technique based on microwave coherent detection of SpBS intensity and frequency shift has also been proposed [38], where measurement accuracy is affected by low strain dependence of the SpRS intensity. It is worth noting that methods using hybrid techniques or birefringence allow better discrimination compared to others, for instance ones using different acoustic optical or acoustic modes, since they involve higher decoupling between the variations of BFS due to temperature and strain.

Hybrid schemes combining RDTS and BOTDA using a single source and sensing fiber have also been used to address this issue and optical pulse coding has been used to provide SNR improvement in both schemes [39, 40]. Among others, a hybrid Raman/BOTDA for simultaneous measurement of strain and temperature has been demonstrated using cyclic coding [41, 42]. The hybrid configuration, which is shown in Figure 17, is the same as the one in Figure 15, except that a Raman filter separates the Stokes and anti-Stokes signal of the SpRS at the receiver, which are detected by APDs, and the remaining signal containing the SBS signal is detected using a pin photodiode. Using a 511-bit cyclic code enables measurement at a temperature resolution of ~3.4°C, which also addresses the cross-sensitivity, yielding a strain resolution of 80 με at the end of a 10-km fiber with a spatial resolution of 1 m.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.