The most general model of elliptical birefringence in an optical fiber has been developed for a steady-state and transient stimulated Brillouin scattering interaction. The impact of the elliptical birefringence is to induce a Brillouin frequency shift and distort the Brillouin spectrum—which varies with different light polarizations and pulsewidths. The model investigates the effects of birefringence and the corresponding evolution of spectral distortion effects along the fiber, providing a valuable prediction tool for distributed sensing applications.
© 2014 Chinese Laser Press
Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light [1,2]. In optical fibers, the birefringence effect is detrimental for a variety of reasons. The Brillouin gain depends on the state of polarization (SOP) in the fiber [3–6], and unintended birefringence causes the polarization of the optical field to change during propagation through the fiber, which induces power fluctuations , especially in the stimulated Brillouin scattering (SBS) process . The local refractive index changes associated with density fluctuations cause the Brillouin spectrum shape to change and the Brillouin frequency peak position to shift . Such induced Brillouin peak shift and spectral distortions may be attributed to local temperature or strain change, and cause error in distributed temperature and strain measurement [7–9]. In , the Brillouin gain spectra were shown to be strongly influenced by thermal stress.
It is important to devise a comprehensive characterization of the SBS process in the presence of birefringence in an optical fiber, so that a prediction of the Brillouin frequency shift and birefringence variation over different sensing lengths can be made. All previous theoretical works [3–5,11–14] are related to the Brillouin gain variation due to the SOP change. Although the effects of fiber birefringence on Brillouin frequency shift and linewidth have been studied experimentally , no report has been made regarding the Brillouin frequency shift associated with the SOP and birefringence change in relation to the fiber position.
A quantitative study of the fiber birefringence versus frequency shift is essential in finding the maximum impact of fiber birefringence on the measurement precision of temperature and strain, for distributed Brillouin optical time domain analysis (BOTDA) or Brillouin optical time domain reflectometry (BOTDR) sensors, as it will be helpful in designing the best suitable fiber for BOTDA or BOTDR applications as well as optimizing system design.
Among early works that investigate the polarization effects on SBS in optical fibers, Refs. [3,5,11,12] showed that the Stokes gain is strongly dependent on polarization effects, and in  this theoretical work was experimentally confirmed. Reference  examines the applications of optical birefringence in SBS sensing for strain and temperature measurements, while  devises a technique to overcome the sensitivity of pulse delay to polarization perturbations, enhancing SBS slow light delay. In , a vector formalism was used to characterize the effects of birefringence on the SBS interaction. One such effect was signal broadening as a result of polarization effects. However, only linearly polarized (LP) pump and signal waves were investigated in [13,14]. Additionally,  assumed an undepleted pump regime, which has applications only for short fiber lengths, while the BOTDA and BOTDR often operate on long fiber distances; the convolution of the birefringence and depletion would induce much larger distortion on the Brillouin spectrum, and hence lower the temperature and strain resolution. Thus the undepleted and LP models are not adequate to address real problems.
The above-mentioned works [5,11,12,14,16], however, treat a steady-state SBS system in which both the pump wave (PW) and the Stokes wave (SW) are continuous. Additionally, none of these references investigated the effect of birefringence and polarization effects on the spectral distortion—namely Brillouin frequency shift—of pulses undergoing SBS. In , though pulse length was taken into account, Brillouin spectrum distortion was not. More importantly, the impact of the nonlinear effect under different pump powers convoluted with fiber birefringence and its impact on the Brillouin spectrum shape and peak shift have not been examined yet. An extension of  was published in the work , whereby signal pulses were taken into consideration and pulse distortion was observed. However, this work still did not take into consideration the most general case of birefringence, which is elliptical birefringence.
It is important to investigate a more accurate model of the polarization-dependent SBS interaction, which includes the case of elliptical birefringence. The model presented in this manuscript describes equations that can be thought of as the most comprehensive SBS equations considering the birefringence effects in an optical fiber. Besides being able to accommodate the most general case of elliptical birefringence, the effects of polarization mode dispersion (PMD), polarization dispersion loss (PDL), phonon resonance structures, pulse length, and the overall attenuation of the fiber have been taken into account.
This manuscript will introduce several important applications of birefringence in optical fibers, relating to telecommunications and fiber sensing. The effect of the spectral shift due to increased birefringence is investigated. Additionally, spectral distortion due to various degrees of birefringence will be investigated for steady-state and transient pulsed regimes. Namely, the steady-state model presented for LP light will be shown to be a valuable measure of the experimentally realistic case of nonideal LP light in optical fibers. The degree of spectral distortion may be used as an indication of the quality of linear polarization during the SBS interaction or as a measure of power leaking between the fast and slow modes. Furthermore, increased power leaking between the fast and slow modes for LP can be used to create a regime that is more favorable for sensing applications related to SBS. In the pulsed regime, spectral broadening and depletion of the Stokes spectrum will be observed as a result of increased birefringence. Spectral distortion is detrimental for fiber sensing and telecommunications; hence methods of minimizing this effect are important for these applications.
Additionally, the effects of various elliptical polarizations on output spectral shape will be investigated for the steady-state model, including spectral hole burning effects and spectral broadening. Methods for maintaining a pulse fidelity and full width at half-maximum (FWHM), as compared to nonpolarized light in a nonbirefringent fiber, will be proposed.
The process of SBS has been studied in a birefringent polarization-maintaining optical fiber (PMF) with a core radius of 4.1 μm. The configuration is composed of a PW launched into one end, and a SW launched into the other end. Both the PW and SW have and eigenpolarization components. The schematic arrangement is shown in Fig. 1.
In the slowly varying amplitude approximation, the interaction between the PW, the SW, and the corresponding acoustic wave (AW), as shown in Fig. 1, is described by the system of Eqs. (1.1)–(1.4). Other than the usual slowly varying amplitude approximation, the only additional approximation in establishing the following equations is the assumption that the phonon fields are established almost simultaneously—which is not a bad approximation for the majority of practical cases [1,18,19]. The derivation of the system of Eqs. (1.1)–(1.4) is shown in Appendix A:6], where is the angular frequency of the PW, and is the angular frequency of the SW. and , and and , are the indices of refraction associated with the principal axes of the PW and SW, respectively. is the angular frequency of the AW caused by the interaction of the PW and SW. and , and and , are the complex amplitudes of the PW and the SW, respectively. is the speed of light, is the mean density of the fiber, is the electrostrictive constant, is the coordinate along the fiber, is the speed of sound in the fiber, is the Brillouin linewidth, and finally, , , , and represent the fiber attenuations of the principal axes of polarization of the two interacting waves.
, , , and represent the Stokes vectors in the Poincaré sphere polarization representation [20,21], and are used to define the polarization of the propagating lights [2,20]. Some extra relations of importance are the following: for forward propagating light (PW) and for backward propagating light (SW). Furthermore, it is worthwhile to emphasize that in the above system of Eqs. (1.1)–(1.4), describes the principal SOP vector for the forward propagating light, and describes the principal SOP vector of the backward propagating light. In this case, if is defined as , then it must follow that , and neglecting the dispersion effect, and , where , , and are the Stokes vector components—the third component, , describing circular birefringence—and are normalized such that . From this arrangement it is apparent that for fibers having elliptical birefringence (, the most general beating situation will be excited; i.e., there will exist four acoustic resonances.
Lastly, the remaining simplifications were introduced to the system of Eqs. (1.1)–(1.4) to make the birefringence effect more explicit. This can be justified by the fact that birefringence and polarization-dependent loss are both small. It is defined , , , and , , , .
In the above arrangement, the PW input parameters are known only at the beginning of the fiber, i.e., at . Correspondingly, the SW input parameters are known only at the end of the fiber, i.e., at , where is the length of the fiber. Therefore, the boundary conditions for the system of Eqs. (1.1)–(1.4) are similar to previously studied configurations with one pulse [6,22]. The conditions for two pulses are as follows:1) becomes 3.1)–(3.4), we have employed the dimensionless length variable , and the dimensionless time variable , where is the transit time, is the average index of refraction, and is the radius of the fiber core. The dimensionless intensity variables are defined as the ratio of powers , , , and . Additionally, , , , and are the dimensionless loss terms, defined as , , , and . The form factor component of the coefficients is defined as , where and . The method of characteristics was employed as in [2,23–25], and the following change of variables was used, where the approximation was used: 4.1)–(4.8): 3.1)–(3.4) of counterpropagating waves into the system of Eqs. (7.1)–(7.4) of copropagating waves. Consequently, we are able to set the following initial conditions:
Though there exist many numerical methods of solution for SBS equations [23,26–31], the fourth-order Runge–Kutta method (RK4) was used to numerically solve the system of Eqs. (7.1)–(7.4), and was chosen for its stability and relatively large step size. Details of this numerical method of solution are summarized in Appendix A.
3. RESULTS AND DISCUSSION
Output spectra were calculated by detuning the Stokes frequency, . Output powers were calculated as , , , and , and the total powers of the PW and the SW were calculated to be and , respectively. Also, the attenuation in the fiber has been approximated as , and the following parameters of the fiber were used: , , , , , and . The following indices of refraction will be used to represent the elliptical birefringence, , of , , , and , respectively:
A. Spectral Shift
The appearance of a fast and slow axis results in two optical modes in the fiber with different SBS frequency shifts, causing a mismatch in the corresponding momentum vectors of the AWs, thereby making it impossible for both principal axes to be resonant with the acoustic phonons. The mismatch in phonon resonance causes a Brillouin shift, , and the larger the birefringence, the larger the Brillouin shift.
Figure 2 shows the magnitude of the Brillouin shift in the output PW and SW spectra.3 summarizes the simulated values of , in comparison to the beat length for LP light. Two cases were compared for pump and Stokes input powers of 1 mW: (a) the input powers of the and pump wave components of the PW and the SW were taken to be unequal, and (b) the input powers of the and components of the PW and the SW were taken to be equal. From Fig. 3, it is apparent that the degree of birefringence has a nonlinear effect on the Brillouin shift ; in particular, the larger the birefringence, the larger the shift. This spectral shift can in turn be used to quantify the birefringence of the optical fiber upon measurement of the output signal. In addition, the spectral shift appears to be more prominent for unequally balanced input powers of the and components of the PW and the SW, as compared to the case of equally balanced powers.
B. Spectral Distortion
1. Linear Polarization
The following simulations were performed for LP light. Figure 4 shows the output spectra of the PW and the SW, as well as close-ups of the spectral tips—both of which are linearly horizontally polarized (LHP) in one simulation, and linearly vertically polarized (LVP) in another, the resulting spectra being identical for both LHP and LVP light.
It is often experimentally not possible to achieve a 100% separation of power between the slow and fast modes of the signal. There is power leakage between the and axes , especially when additional optical components, such a polarization controllers or scramblers [33,34], are used. Hence, for a 11 mW signal, it is a realistic assumption that as much as 1 mW of power could be present in the vertical mode of the signals for LHP light. Likewise, for LVP lights, it is also experimentally plausible to have as much as 1 mW of power in the horizontal mode of the signals.
The shape of the output spectra in Fig. 4 can be explained by the modified interaction between the PW and the SW during LP. Referring to the system of Eqs. (7.1)–(7.4) and coefficients (4.1)–(4.3), (4.8), we see that the coefficients , , , and are reduced to zero for both LHP and LVP light. As a result, the system [Eq. (7) is broken up into two smaller systems consisting of (a) Eqs. (7.1) and (7.3) and (b) Eqs. (7.2) and (7.4), respectively, which are independent of each other. The first system describes the interaction between the horizontal components ( components) of the PW and the SW, and the second system describes the interaction between the vertical components ( components) of the PW and the SW. In other words, interactions along the and components become independent from each other, which is to be expected during linear polarization.
In Fig. 4, which shows the output pump and Stokes spectra of LHP light for various pulse lengths, the little “blip” at the bottom/top of the spectra is the -component contribution, while the “general” shape of the spectra is a result of the -component contribution. Since the power in the horizontal mode is large for LHP light, the resultant depletion of the PW and amplification of the SW are also large, leading to the general trend of the output spectra in Fig. 4. This is the spectral shape that would be expected for an idealized LHP light (or perhaps light with no consideration for polarization at all). However, the LHP is not ideal, and there exists a small interaction in the vertical component. Since the power along the axis is small, the resultant contribution is also small. As can be seen from Fig. 4, the smaller the power in the component, the smaller the contribution, and the weaker the spectral distortion of the graph. In other words, increasing powers in the components increase the degree of spectral distortion. For , the spectral “blip” is much larger than for the case in which , which is practically nonexistent. However, while a smaller power leakage causes a smaller spectral distortion or “blip”, it yields output spectra with a larger depletion, or “flat-top,” than in the case of larger power leakage. For the steady-state regime, shown in Figs. 4(a) and 4(b), where both the PW and the SW are continuous waves, the “flat-top” effect is more prominent, and the spectral “blip” is less pronounced, as compared to shorter pulse lengths of 240 and 79 ns, shown in Figs. 4(c) and 4(d) and Figs. 4(e) and 4(f), respectively. With decreasing temporal pulsewidth, the power leakage causes the distorted spectral tip to be sharper and more pronounced, as compared to longer pulses.
For sensing applications, it is detrimental to have a depleted spectrum [7–9], since depletion and spectral flattening make it difficult to accurately measure the center frequency of the spectrum. Although the distorted spectra with smaller depletion, obtained from the steady-state interaction of the PW and the SW, may not have a sufficiently prominent spectral tip for sensing applications, it is an improvement nonetheless, as compared to the case of larger depletion, in which the “flat-top” spans an even larger frequency range.
As shown in Figs. 4(e) and 4(f), a shorter pulse length of 79 ns undergoing spectral leakage between the fast and slow modes has a sufficiently prominent spectral tip for measurement in sensing applications. As such, by using the distortion effects caused by birefringence to our advantage, it is possible to provide a regime that is favorable for sensing applications related to SBS, by increasing the power leakage between the fast and slow modes, as well as decreasing the pulse length during LP.
In the case of LVP, also shown in Fig. 4, the roles of the and components are reversed, but the interaction is identical. For this reason, the two cases of LHP and LVP light, shown in Fig. 4, give identical results, as in either case, the interaction along the slow or fast axes remains the same.
Also, in Fig. 4, the output spectra of LP light for all pulse lengths are shown in comparison to light without a dependence on polarization, depicted by the dashed line—namely, light having a Stokes vector , which has been taken to be a first-order approximation to truly unpolarized light. The spectra of the LP are spread out in the frequency domain, as compared to nonpolarized light in a nonbirefringent fiber. This effect can be explained by the nature of the LP interaction between the PW and the SW. As discussed above, when LP is not ideal, there exists a small independent interaction along either the fast or the slow mode, contributing to the spectral distortion. The interaction along the main axis of polarization, however, contributes the most to the output spectral shape of the light—the SOP of the counterpropagating waves being lined up for maximum gain [3,11,12,14,16,17]. Additionally, since this interaction is strong, and the input powers along the main mode are comparatively large, the output light experiences a significant gain/depletion, causing the spectrum itself to be larger and inherently spread out . This effect does not present itself for unpolarized light, where there cannot be polarization alignment for maximum gain. All four lights interact with each other, and as a result, the output spectrum is smaller, which contributes to less spectral spreading in the frequency domain.
As a result, the model presented for LP light proves to be a valuable measure of the experimentally realistic case of nonideal LP light in optical fibers, or a measure of power leakage between the fast and slow modes. The degree of spectral distortion may be used as an indication of the quality of LP during the SBS interaction.
2. Elliptical Polarization for Steady-State Interaction
In this section, the continuous PW and SW were simulated to have several elliptical polarizations: Random 1 (0.1, 0.9, 0.42), Random 2 (0.3, 0.7, 0.65), Random 3 (0.58, 0.58, 0.58), and Random 4 (0.1, 0.9, 0.42). The effect of elliptical polarization on the spectral shape of the output light was observed for pump and Stokes input powers: (a) below the Brillouin threshold and (b) above the Brillouin threshold [1,2].
In Fig. 5, the individual and components of the SW have been plotted. Moderate powers have been chosen for various degrees of birefringence: , , and . It can be seen that the spectral distortion that results due to the birefringence is more prominent for higher degrees of birefringence () as compared to lower degrees of birefringence ().
In the most general case of elliptical birefringence, there are four running AWs, each having its own resonance frequency. As a result, each of the fast and slow modes has its own resonant frequency, which is the cause of the multiple peaks on the output spectra in Fig. 5.
In Figs. 6 and 7, input powers have been taken beyond the Brillouin threshold for various elliptical polarizations. It can be seen that certain polarizations cause a kind of spectral hole burning effect [35,36] to take place in the output spectrum, while others affect the spectral shape negligibly.
In Fig. 6, for example, the spectral hole burning effect can be explained as a result of a very strong interaction along one mode of the optical fiber, and weak interactions along the other modes for the polarization Random 4 (0.1, 0.9, 0.42). In Fig. 6, since the input power of the Stokes component is so large (80 mW), and the corresponding vertical polarization component is also large (0.9), it acts to quickly deplete power from the other modes, and since, as mentioned previously, there is no single resonant frequency, it does so along multiple resonant frequencies. For polarizations Random 1 (0.6, 0.25, 0.76), Random 2 (0.3, 0.7, 0.65), and Random 3 (0.3, 0.7, 0.65), the combination of the relatively weak vertical polarization component and the strong circular polarization component almost eliminates the effect altogether. Polarizations Random 1 and Random 2 are nearly identical to the nonpolarized light; hence these polarizations can be used to maintain the fidelity of the spectral shape to be as close as possible to nonpolarized light, which is unaffected by birefringence. Additionally, the effects of distortion on the FWHM of the output pulse can be minimized, which has numerous applications in communications and data transmission.
A similar general spectral broadening effect can be observed in Fig. 7, though because of the different power distribution among the modes there is no longer a spectral hole burning effect. In addition, polarizations Random 1, Random 2, and Random 3 are no longer nearly identical to the unpolarized light, due to the different interactions between the four lights caused by varied initial power distributions.
As a result, it has been shown that elliptical birefringence has a prominent effect on the spectral shape of the output PW and SW. In some cases, it causes a spectral hole burning effect as well as spectral spreading, while in other cases it is possible to choose a polarization and power combination to minimize the spectral distortion of the FWHM of the output pulse, and maintain pulse fidelity.
In summary, it has been observed that the degree of birefringence, or beat length, is responsible for an observed Brillouin shift, , in the output PW and SW spectra. Polarization and elliptical birefringence have a prominent effect on the spectral shape of output light—in particular, nonideal LP light causes spectral distortion, which appears in the shape of a “blip” on top of the expected spectral shape, as well as spreading of the spectrum in the frequency domain. As such, a good measure for detecting nonideal LP light has been established. Additionally, due to the smaller spectral depletion of the nonideal LP light, which is preferable for sensing applications, a regime that is preferable for measurement in sensing applications has been proposed. Elliptical polarization and birefringence, for moderate input powers of the PW and the SW, also cause the spectrum to spread out in the frequency domain. For high input powers, a spectral hole burning effect is predicted for certain elliptical polarizations. Other elliptical polarizations provide means for maintaining the FWHM and pulse fidelity of the Brillouin spectral shape—even for powers beyond the Brillouin threshold.
Let us consider the simple case of two propagation constants in the fiber, calling them and for now. If a light with an angular frequency was injected in the fiber, the electric fields could be written in the following form, in terms of their principal axis of polarization:
Taking the steady-state approximation, we have the complex acoustic field amplitude for the simplest case of zero birefringence:6]. Using the approximation , where and are taken to be the averages of and , and and , respectively, it is then possible to have the following approximate complex acoustic field amplitudes,
Paying attention to the original case without birefringence in Eq. (A5), it is possible to generalize our result. Recall that nonbirefringent lights were assumed to undergo ideal power transfer between the PW and the SW, as would be the case when both of these waves are parallel polarized (under the slowly varying amplitudes, and , approximation),A7.1) and (A7.2), is the mean density of the fiber. Generalizing the above system of Eq. (A7) to the case of birefringence we take the following: 1).
2. Fourth-Order Runge–Kutta Method of Solution
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