## Abstract

An extended theory of the mutually modulated cross-gain modulation was developed to make it valid under considerable modulated wavenumber-interaction length product. Parametric analysis of the extended theory shows that, in the modulated probe beam out of the gain medium, the spectral sensitivity of the first harmonic wave is associated with its amplitude. The larger the sensitivity is, the smaller the amplitude will be. An effective method was provided to monitor and achieve high spectral sensitivity at a pre-selected frequency detuning in practice. The validity of the extended theory was confirmed by a sensing experiment in a piece of ∼10-km fiber with 10 kHz modulation frequency, in which a fast light propagation with a sensitivity of 0.592 mrad/kHz was achieved.

© 2015 Optical Society of America

## 1. Introduction

Mutually modulated cross-gain modulation (MMXGM) is a technique which can be applied to the gain medium with both the pump and probe beams modulated at the same frequency. With suitable modulation depth and phase shift, it can be applied to realize not only group velocity control on the modulated beam [1–4], but also optical frequency sensing in the probe beam [5–7]. The sensitivity can be up to around 10 mrad/kHz in stimulated Brillouin scattering (SBS) media [5,6]. In the sensing process, the spectral sensitivity is enhanced in the modulated phase rather than the optical phase [8, 9], which makes it convenient for many optical applications because it does not require an interferometer structure.

Although the theory of MMXGM has been developed and its validity was confirmed experimentally in various applications, current theories and experiments were under a condition of *KL*≪1, where *K* is the modulation wavenumber proportional to the modulation frequency and *L* is the length of the sample. However, MMXGM applied under a high modulation frequency is of great value for its applications. In the application of group velocity control, higher modulation frequency means larger bandwidth, i. e., the modulated pulse width in certain sample could be shorter. In the application of MMXGM frequency sensing, the current method to monitor the sensitivity is to compare the modulated phase difference while changing the frequency of the probe beam. A higher modulation frequency indicates a potential faster measurement. Under the condition of high modulation frequency, applying MMXGM in a long sample such as fiber of several kilometers may result in a *KL* comparable to or even larger than unit. Meanwhile, in our previous work [6], it is found that the maximum sensitivity is dependent on *KL* in a MMXGM frequency sensing process (See Eq. (9) in [6]). A slight change in *KL* may affect the maximum sensitivity significantly even under the condition of *KL*≪1, which shows the importance of *KL*. We also found that as the sensitivity goes higher, the amplitude of the first harmonic wave will be smaller in the calculation (Fig. 2 in [6]). During the process to optimize the sensitivity, it is much easier to monitor the amplitude of the first harmonic wave than to observe the phase shift, especially when the amplification process of MMXGM is not stable, which will be quite useful in the application. However, we cannot get analytic result to support this fact based on the current theory because of simplification based on *KL*≪1. With these demands, MMXGM with considerable *KL* is expected.

In this article, we present an extended theory suitable for considerable *KL* and the result of MMXGM sensing in a ∼ 10-km fiber at a modulation frequency of 10 kHz, i. e., *KL*≃3. Theoretical analysis shows that the sensitivity is associated with the amplitude of the first harmonic wave. Under most conditions, the larger the sensitivity is, the smaller the amplitude of the first harmonic wave will be, which is also supported by our experimental results with a sensitivity of 0.592 mrad/kHz. To achieve high sensitivity in application, band-pass filter is necessary to block high order harmonic wave. The research will be helpful for the further applications of MMXGM.

## 2. Principle and theory

Figure 1 shows the conceptional scheme of MMXGM technique. A pump beam with an angular frequency *ω _{p}* and a probe beam with an angular frequency

*ω*are launched in the −z and +z directions into a gain medium with a length

_{s}*L*, respectively. Both the pump and the probe beams are modulated at the same frequency

*f*. Due to the energy and phase couplings between the pump beam and the probe beam, the probe beam will undergo a resonance amplification at a frequency deviated from that of the pump beam by

_{mod}*f*. Normally,

_{resonance}*f*is much larger than

_{resonance}*f*. Under such conditions, major fourier components of the modulated probe beam are selectively amplified. This will introduce higher order harmonic waves into the output probe signal, in which the first harmonic wave gets a nonlinear phase change that is extremely sensitive to the frequency change of the probe beam.

_{mod}The theory and principle of laser frequency sensing based on SBS MMXGM have been previously derived in [3] for the specific case when the modulation phase shift *φ*_{2} between the pump and the probe at the front facet of the gain medium (see Fig. 1) is equal to *π*. The latest work [6] discussed the situation when *φ*_{2} is slightly deviated from *π* and found it will lead to an enhancement of the sensitivity. However, both theories are based on the approximation *KL* << 1, where *K* := Ω*n/c* is the modulation wavenumber with *n* and Ω = 2*πf _{mod}* being the refractive index and the modulation angular frequency, respectively. Besides the SBS, the technique can also be applied to the other gain mechanisms, for example, nondegenerate two-wave coupling and semiconductor optical amplifier [7]. In the following, we will give an outline of the primary theory of MMXGM and then extend the theory at the conditions with considerable

*KL*and arbitrary

*φ*

_{2}.

The same as the theory derived in [3], a pump beam of a modulation depth *α* is given by

*β*. At the front facet (

*z*= 0) of the gain medium, the intensity

*I*

_{2}(

*z*,

*t*) of the modulated probe beam is Assuming the linear phases and group velocities of the modulated pump and probe beams are the same, which is a good approximation in the weak dispersion medium, the amplified probe beam at the output facet of the gain medium can be given as [3]

*G*depends on the frequency detuning

*δω*/2

*π*of the probe beam from the resonance frequency

*f*In Eq. (4),

_{resonance}*G*

_{0}is the resonance gain at the resonance frequency

*f*and

_{resonance}*f*(

*δω*) is the gain function. Note that

*f*(

*δω*) could be of different formula according to different gain mechanisms, but its value is always within [0, 1] with

*f*(

*δω*= 0) = 1.

For the case of amplified probe beam, it can be expressed equivalently as a sum of harmonics

*i*and

_{m}*θ*(

_{m}*m*≥ 1) are the amplitude and the phase of the

*m*th order harmonic wave, which are functions of parameters

*K*,

*L*,

*α*,

*β*,

*G*and

*φ*

_{2}.

At the same time, Eq. (3) can also be expanded as

Since the high sensitivity frequency sensing and group velocity control utilize the total phase shift *θ*_{1} of the first harmonic wave, by comparing Eq. (5) to Eq. (6), we get

*i*

_{1}is

Note that the phase shift *θ*_{1} in Eq. (7) actually includes both the linear phase shift *KL* originated from the propagation of the light inside the sample and the nonlinear phase shift analyzed in the previous works [3,5,6]. When there is a change in *δω*, the spectral sensitivity *S* depends directly on the phase change in the first harmonic wave:

*Ĝ*:=

*αG*sinc(

*KL*)/

*β*, Eqs. (8) and (9) can be simplified as

*KL*≪ 1, Eq. (11) is the same as the sensitivity derived in previous works [3, 5, 6] when

*φ*

_{2}=

*π*and

*Ĝ*≈ 1.

The above results (Eqs. (7) – (11)) are directly derived from the
expression of the output probe beam (Eq.
(3)) without any further approximation. There is no such limitation as
*KL* ≪ 1, and therefore they are valid in a long sample
at a high modulation frequency satisfying the condition
*f _{mod}* ≪

*f*.

_{resonance}## 3. Parametric analysis on spectral sensitivity *S*

In this section, the parametric relationships in the MMXGM sensing process with large *KL* are discussed in detail based on the extended theory. In the sensing process, sensitivity *S* is an important parameter; at the same time, the group velocity could be optimized at the place where *S* ≠ 0.

First of all, we analyze the sensing process based on Eq. (11) to make its parametric relationship more clear. One notes that the right side of Eq. (11) can be divided into two parts. The value of *dG/d*(*δω*) is determined by the gain of the medium. Once the gain medium is in a stable environment, i. e., *f*(*δω*) in Eq. (4) is not changing, *dG/d*(*δω*) will also not change with the probe beam being at a certain frequency difference *δω* from the pump beam. The rest part on the right side of Eq. (11) actually includes all the key parameters to control the spectral sensitivity, i. e.,

*Ĝ*> 0. When both

*Ĝ*≈ −cos(

*KL*−

*φ*

_{2}) and

*KL*−

*φ*

_{2}≈ (2

*j*+ 1)

*π*, where

*j*is integer, are fulfilled in the system, the sensitivity could be extremely high. Note the right term of Eq. (12) can be either positive or negative (corresponding to fast or slow light, respectively), which is controlled by the value of

*KL*−

*φ*

_{2}. The corresponding sensitivity is determined by its absolute value and its sign inherits from phase shift. Under a very specific condition when

*KL*=

*jπ*, the phase shift

*θ*

_{1}will not change at all and

*Ĝ*is always zero, as seen from Eq. (7) and the definition of

*Ĝ*, and we know from Eq. (12) that the sensitivity will always be zero.

Equation (12) also determines the position of the sensing window, inside which the sensitivity is high and obvious phase change Δ*θ*_{1} can be observed when the frequency of the probe beam is changed. The position of the sensing window is defined by the frequency detuning of the maximum sensitivity from the resonance gain frequency. Under a certain experimental configuration with given *KL* − *φ*_{2}, i. e., with given sample length, modulation frequency and initial modulation phase shift, the sensing window is located at position where *Ĝ* = −cos(*KL* − *φ*_{2}). In the gain medium described by Eq. (4), the position of the sensing window is actually determined by the value of *Ĝ*_{0} as reported in [6].

In principle, the sensitivity can be as large as we expected but the measured sensitivity in the experiment is usually smaller than that calculated. This is because the measured sensitivity in the experiment is extracted from two measurements on the modulated phase with a small frequency difference in the probe beam, and it is given as

where*S*is the sensitivity achieved in the experiment, Δ

_{measurement}*θ*

_{1}and Δ(

*δω*) are the phase difference and the angular frequency difference between two measurements, respectively. It is obvious from Eq. (4) that

*Ĝ*:=

*Gα*sinc(

*KL*)/

*β*will be different in the two measurements. Since

*Ĝ*is one of the key parameters to determine the sensing window and the sensitivity, therefore the sensitivity will also be different in the two measurements. Only when Δ(

*δω*) is small enough so that the difference mentioned above can be neglected, the profile of the sensing window can then be properly measured with negligible deformation. Otherwise, the measured sensitivity would be smaller than the one calculated from Eq. (11).

The high sensitivity brings about another problem in practical applications. Because the phase shift *θ*_{1} is limited to be less than *π*, high sensitivity would result in a very narrow sensing window. For example, in SBS MMXGM sensing inside a commercial fiber, the width of the sensing window could be only around 1 MHz. Therefore, it is even more difficult to locate the position of maximum sensitivity inside the sensing window, especially when the gain system is not stable so that the sensing window will move accordingly. Although measurement and monitor on the sensitivity through the phase shift is easy to understand, it is hard to operate in practice. Fortunately, by comparing Eq. (10) with Eq. (11), one can obtain the relationship between *S* and *i*_{1}

As mentioned above, the position of the sensing window, i. e., the position of the maximum spectral sensitivity, is determined by *Ĝ*, which, in general, should be determined before the MMXGM technique is applied. It is obviously seen from Eq. (8) that, by sweeping *α* to change *Ĝ*, when the maximum sensitivity is right at the selected frequency detuning, *i*_{1} will be the minimum. Note that *α* can be adjusted within [0, 1]. There may be such situations that the maximum sensitivity is not in the range of adjustment and the system needs to reset parameters by adjusting *β* or *G*_{0}. Once *Ĝ* is selected, suitable *φ*_{2} will result in desired sensitivity. Since *S* is inversely proportional to (*i*_{1}/*i _{DC}*)

^{2}at certain frequency detuning as shown in Eq. (14), one can adjust the parameter

*φ*

_{2}to decrease the amplitude of the first harmonic wave in the MMXGM process and therefore to increase the sensitivity.

## 4. Calculation and experiment

The above discussions actually show us how to achieve maximum spectral sensitivity (how to control group velocity) at a selected frequency detuning in the experiments and practical applications. The conclusion is also supported by previously published results. In the SBS process of a chalcogenide waveguide, the relationship between *S* and *i*_{1} has been calculated based on Eq. (3) and shown in Fig. 2 of [6]. In this section, further calculation of the spectral sensitivity and the output waveforms of probe beam were presented to make it more clear. Meanwhile, a SBS MMXGM sensing experiment was realized in a ∼ 10 km fiber with a modulated frequency 10 kHz (*KL* ≈ 3), which confirms that it is possible to realize MMXGM with large *KL*.

Figure 2 presents the calculated sensitivity based on Eq. (13) in a 3-km fiber with SBS MMXGM sensing at different angular frequency step Δ(*δω*). The parameters used in the calculations are *n* = 1.47, *L* = 3.0 km, *f _{mod}* = 1.0 kHz,

*Ĝ*

_{0}= 2,

*φ*

_{2}=

*π*+ 0.09, and the Brillouin linewidth of the fiber Γ

*= 2*

_{B}*π*× 20 × 10

^{6}rad/s, which is the same as the first sensing experiment in fiber [5] except that

*φ*

_{2}is changed a little to enhance sensitivity. It is clear that, as Δ(

*δω*)/(2

*π*) is increasing from 20 kHz to 200 kHz, the maximum spectral sensitivity calculated based on Eq. (13) is decreasing from −0.039 rad/kHz to −0.010 rad/kHz. The minus sign of the sensitivity only means that the sensing process is a slow-light one [6].

As discussed above, the higher the sensitivity is, the smaller the *i*_{1} is. When the sensitivity is very high, *i*_{1} will be so small that it may be comparable to *i*_{2} and the waveform of the output probe beam distorts significantly. A SBS gain process in fiber was calculated with
${I}_{2}^{0}=1\hspace{0.17em}\hspace{0.17em}\text{W}/{\text{m}}^{2}$, *n* = 1.47, *L* = 10.0 km, *f _{mod}* = 10.0 kHz,

*Ĝ*

_{0}= 2,

*φ*

_{2}= −0.062, and the Brillouin linewidth of the fiber Γ

*= 2*

_{B}*π*× 20 × 10

^{6}rad/s. The waveform evolution under different detunings at high sensitivity is presented in Fig. 3, where Figs. 3(a) and 3(b) show the output waveform evolution of the AC part and the first harmonic wave at different detunings, respectively, and Figs. 3(c) and 3(d) show the output waveform of the AC part and the first harmonic wave when

*δω*/2

*π*= 10 MHz, respectively. It is seen that the second harmonic wave is obvious in Figs. 3(a) and 3(c). It is clear that, as

*δω*changes in the probe beam,

*θ*

_{1}varies fast when

*i*

_{1}is small. At the same time, the DC part of the probe beam was around 2.7 W/m

^{2}, which is at least about 10

^{4}larger than the AC part.

To verify the extended theory, a MMXGM sensing experiment with *KL* > 1 was carried out. In the experiment, a simple pump and probe configuration was adopted without bandpass filter as shown in Fig. 4, where EDFA was the erbium doped fiber amplifier, EOFL1 and EOFL2, driven by function generators FG1 and FG2, respectively, were intensity modulators to modulate the pump and probe beams, EOFH driven by a signal generator SG was the intensity modulator to generate the frequency-shifted probe beam, C1, C2 and C3 were the circulators, OSA was an optical spectrum analyzer. The pump beam was modulated at 10 kHz and sent into a ∼ 10-km fiber loop, while the probe beam was modulated at the same frequency and sent into the other end of fiber, too. After filtered by the fiber Bragg grating FBG, the output probe beam with SBS gain was split by a coupler and sent into optic-electronic detector D and OSA, respectively. After obvious SBS gain was observed in the OSA (more than 10 dB in our experiment), we optimized the parameters as discussed above, *i*_{1} observed in the oscilloscope (OS) was minimized and the second harmonic wave could be observed when the output of SG was around 10.843 GHz. As the frequency difference (*δω*)/2*π* was swept from 10.841 GHz to 10.845 GHz and approached to the Brillouin resonance frequency around 10.85 GHz, the waveform evolution of the AC part in the probe beam was measured and shown in Fig. 5(a). The evolution of the first harmonic wave was achieved in computer by band pass filtering (FFT, 9700 Hz −10300 Hz), which is presented in Fig. 5(b). Note the waveform are all presented in a waterfall-style with reference level of its amplitude shifted. From Fig. 5(a), it is clear that the group velocity shows fast light property and the sensitivity calculated by the waveform at 10.842 GHz and 10.843 GHz is 0.592 mrad/kHz. The waveform is distorted significantly, mainly due to the noise in the measurements and high order harmonic waves especially the second harmonic wave. The amplitudes at the two frequencies are the smallest among measured waveforms, as it has been analyzed in the extended theory. The experiment shows that, under the condition of large *KL*, MMXGM technique could also be applied to realize group velocity control and frequency sensing. With an electronic bandpass filter, the amplitude of *i*_{1} is more easy to observe and higher sensitivity could be achieved in practice.

## 5. Conclusion

As a conclusion, an extended theory of MMXGM for considerable *KL* was developed. Both theory and experiment in a ∼10-km fiber at a modulation frequency of 10 kHz show that MMXGM could be applied under more large modulation frequency and therefore provides larger bandwidth in group velocity control and possible faster measurement in the frequency sensing. In the theory, we also associated the sensitivity with the amplitude of the first harmonic wave in the MMXGM sensing process and analyzed the influence of the key parameters. In general, the larger the sensitivity is, the smaller the amplitude of the first harmonic wave is. It shows us how to control the position of the sensing window and the desirable sensitivity, which is supported by the sensing experiments of large *KL*. These results would be helpful for further applications of MMXGM.

## Acknowledgments

This work is financially supported by the 973 Programs ( 2013CB328702, 2013CB632703), the CNKBRSF ( 2011CB922003), the NSFC ( 11174153, 11374165, 11404263 and 61475077), the 111 Project ( B07013), the Natural Science Foundation of Tianjin ( 12JCQNJC00900) and PCSIRT ( IRT0149).

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