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Abstract
We present a new all-optical method for the magnification of small-intensity fluctuations using the nonlinear Kerr effect. A fluctuation of interest is impressed onto a sinusoidally modulated optical signals (SMOS) and spectral sidebands are generated as the SMOS experiences self-phase modulation in a nonlinear medium. Magnification of these temporal variation is obtained by filtering one of the sidebands. For small fluctuations, the amount of magnification obtained is proportional to (2m + 1), with m being the sideband order. This technique enhances fiber-based point sensor capabilities by bringing signals originally too small to be detected into the detection range of photodetectors.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
Benoit Vanus, Chams Baker, Liang Chen, and Xiaoyi Bao, "All-optical intensity fluctuation magnification using Kerr effect: erratum," Opt. Express 29, 38082-38083 (2021)https://opg.optica.org/oe/abstract.cfm?uri=oe-29-23-38082
1. Introduction
Most optical intensity based sensors, such as a Mach-Zehnder interferometer [1–4] exhibit a sinusoidal power dependance to a parameter of interest, such as strain or temperature, as illustrated in Fig. 1(a). If the intensity fluctuation that results from the measured parameter variation is too weak to be detected, illustrated as a sinusoidal signal in Fig. 1(b), an amplifier, such as an erbium-doped fiber amplifier, is utilized to amplify the intensity of the fluctuation to a detectable level. However, this leads to the amplification of both the useful signal and its offset, the latter is defined as the optical power level at the base of the signal of interest, as illustrated in Fig. 2. This can lead to a saturation of the photodetector because the intensity becomes higher than the saturation level of the photodetector, as illustrated in Fig. 1(c) where the dashed line represents the saturation level. Hence there exist a maximum amplification beyond which the signal can not be detected. To extract a small fluctuation and remain below the photodetector saturation limit, a magnification of the signal of interest must be combined with a reduction of its offset, as presented in Fig. 1(d). A solution to this issue can be found using nonlinear optical processes as some allow for optical signal reshaping and reamplification based on self-phase modulation (SPM) [5–8] and for general all-optical signal processing when including cross-phase modulation, three and four wave mixing processes [9].
Fig. 1. Illustration of the amplification and magnification of small intensity fluctuations. $\epsilon$: Strain, P: Power, PD: Photodetector, T: Temperature, V: Voltage. a) Optical sensor operating at the quadrature point, b) The original signal power with respect to the PD saturation level, c) The amplified signal is above the saturation level and can not be detected, d) The magnified signal is below the saturation level and can be detected.
Fig. 2. Illustration of the signal parameters, defining the maximum and minimum powers as well as the signal offset power.
In this paper, we propose and demonstrate the use of SPM to magnify a signal intensity fluctuation and reduce its offset power to prevent photodetector saturation. First, we present the theory behind optical signal magnification and define the signal contrast. Then, we predict the magnification factor of the signal contrast for small amplitude signals. Finally, we present an experimental setup generating all-optical small signals magnification and show a close agreement between theoretical predictions and experimental results.
2. Magnification of optical signals
It has been demonstrated that spectral sidebands are generated through SPM if a sinusoidally modulated optical signal (SMOS) featuring a high peak power undergoes the nonlinear Kerr effect [10,11]. All-optical signal magnification using the nonlinear Kerr effect is achieved by operating at power levels exhibiting an exponential relationship between powers of the input signals propagating through the Kerr medium and the generated sidebands. By adjusting the peak power of the signal sent into the Kerr medium, any deviations from the original power profile will get magnified. The maximum power of the signal $P_{max}$ is defined as the peak power of the signal, the minimum power $P_{min}$ as the power at the troughs which also corresponds to the offset power of the signal $P_{offset}$ , and the signal’s magnitude $P_{signal}$ is defined as the difference $P_{max}-P_{min}$, as illustrated in Fig. 2. The signal contrast at the input of the Kerr medium, $\rho _{in}$, is expressed as :
Based on Fig. 2, the total power is described as the sum of the offset power and the signal power: $P_{tot}=P_{offset}+P_{signal}$. Therefore, $\rho _{in}$ can be expressed as: Figure 3 shows a linear dependance at small nonlinear phase shift $\phi _{SPM}=\gamma P_{p}L$, in the logarithmic scale, of each SPM-generated sideband relative output power variation ($P^{(m)}/P^{(0)}$), where $m$ is the sideband order, $\gamma$ is the waveguide nonlinearity parameter, $P_{p}$ is the signal peak power and $L$ is the length of the nonlinear medium. In this regime, any fluctuation in the initial peak power leads to a variation in $\phi _{SPM}$ and thus leads to a magnification of this fluctuation impressed on the power of the sidebands. A linear regression with a slope of $2m$ is fitted to each sideband order and shows good agreement for small signals, as presented in Fig. 3. Therefore we can approximate that, for small power variations, the relative output power variation is proportional to the $2m$ power of the nonlinear phase shift: $P^{(m)}/P^{(0)}\propto \phi _{SPM}^{(2m)}$. Moreover, by operating at a low power regime, it can also be approximated that the input signal power is equivalent to the power of the $0^{th}$ order sideband: $P_{in}\approx P^{(0)}$; therefore, as the input power is proportional to the peak power, $\phi _{SPM}\propto P_{in}$, we have that $P^{(m)}/P_{in}\propto P_{in}^{(2m)}$. Thus the power of the $m^{th}$ order sideband is proportional to the input power raised to the power of $(2m+1)$: The contrast of the $m^{th}$ order sideband is expressed as:Fig. 3. Relative output intensity for each sideband as a function of the nonlinear phase shift ($\phi _{SPM}$) using the equation $P^{(m)}(\phi _{SPM}) = P_0\left [ J_m^2(0.5\phi _{SPM}) + J^2_{m+1}(0.5\phi _{SPM})\right ]$ [11], fitted with a slope proportional to $2m$, with $m$ being the sideband order, $P_0 =P_p/4$ and $P_p$ the input signal’s peak power.
3. Experimental setup and results
Figure 4 presents a schematic of the signal magnification system. A 1550.12 nm light is generated by a distributed continuous wave (CW) feedback laser (AOI DFB-934-BF-10-EC-Fx-Hx-N126) and gets amplified using an erbium-doped fiber amplifier (EDFA, Amonics AEDFA-C-30B). An electro-optic modulator (EOM, OC-192) applies a 100 ns square pulse modulating the CW light with a repetition rate of 100 kHz. This pulse modulation prevents the formation of accumulated Brillouin effect in the Kerr medium which would lead to a decrease of the laser signal power at the original wavelength, hence preventing the SPM process to occur. A second EOM (OC-192) then applies a 4 MHz sinusoidal modulation on the pulse peak power to emulate the power fluctuation to be magnified. Finally a third EOM (OC-192) applies a 9.2 GHz sinusoidal modulation on the overall signal, creating an SMOS. The modulated overall signal is then amplified and filtered by two cascaded fiber Bragg gratings (FBGs) centered at $\lambda _{1}$=1550.058 nm and $\lambda _{2}$= 1550.202 nm. Power is then boosted using a high-power EDFA (Amonics EDFA033) before undergoing a second filtration step. This step allows to further remove any residual carrier contributions and equalize optical paths by using a mirrored version of the initial FBG filter. The light then experiences SPM in a 2-km long dispersion-shifted fiber, with a zero dispersion wavelength of 1550.2 nm. The SPM generates sidebands at the output of the Kerr medium [11] and the first and second order sidebands are respectively extracted using a band-pass filter with a 3 dB bandwidth of 3 GHz (BPF - TFC-C-Band). The input power of the nonlinear medium is adjusted to maximize the contrast of the signal at the output of the Kerr medium. A low-noise photodetector (New Focus 1811- IR DC 125MHz) interfaces an oscilloscope (LeCroy 64Xi-A) and enables the measurement of the magnified signal.
Fig. 4. Schematic of the optical signal magnification experimental setup. BPF: Band-Pass Filter, EDFA: Erbium-Doped Fiber Amplifier, EOM: Electro-Optical Modulator, HP-EDFA: High-Power EDFA, KM: Kerr Medium, OSC: Oscilloscope, PC: Polarization Controller, PD: Photodetector
Using the setup presented in Fig. 4, we first apply a 1V modulation on EOM2 and compare the reference signal, measured at point A in Fig. 4, to the filtered first and second sidebands. Figure 5 presents the normalized original signal power and the normalized power of the first and second order sidebands. We can observe a magnification of the sinusoidal modulation overlaid on top of the pulse, this magnification increases as the order of the sideband increases. The normalized reference is raised to the power of $(2m+1)$ to represent the theoretical prediction of magnification according to Eq. (3). The theoretical calculations of the normalized powers of the first and second order sidebands are presented in dashed lines in Fig. 5 and show good agreement with the experimental results. The discrepancy between the measurement and the prediction appearing in the first sinusoidal peak is explained by the fact that the reference pulse signal does not exhibit a constant peak power, hence when normalized and raised to the $(2m+1)$ power, it leads to amplitude values below unity.
Fig. 5. Experimental measurement and theoretical approximation of a normalized reference signal and its magnification of first and second order.
To prove the $(2m+1)$ magnification ratio of small signals and faithful linear correspondance between input and output signals, the modulation voltage on the EOM2 is varied, which modulates the magnitude of the sinusoidal signal. To measure the ratio between the output signal contrast and the input signal contrast, $\kappa =\rho _{out}/\rho _{in}$, the modulation voltage is reduced from 2V to 50mV by halving the value at each step. Figures 6(a) and 6(b) respectively present the original normalized signal power for a 50 mV modulation voltage and the normalized second order sideband signal power showing a magnification by a factor of 5. Figure 6(c) presents the variation of $\kappa$ as a function of the signal modulation voltage for the first and second order sideband. The contrast between the second trough and the third peak (see Fig. 5) is chosen to calculate the contrast ratio to avoid the leading and falling edge pulse dynamics imposed by the function generator. We observe that the measurements in the small-signal regime are in agreement with the predicted $(2m+1)$ value of amplification given by Eq. (5) as they respectively saturate at values of 3 and 5 for the first and second order sidebands. For $\phi _{SPM}$ values within the linear regression regime in Fig. 3, the magnification value increases with $m$, and the setup can be cascaded to achieve any magnification value. This opens the way to enhance the sensitivity of intensity based sensors as will be demonstrated in future works.
Fig. 6. Measurement of the magnification of small intensity fluctuations. a) Measured normalized original signal with a modulation depth of 50 mV, b) measured normalized $2^{nd}$ order sideband signal, c) ratios between of the output and input signal’s contrast as a function of the modulation depth of the sinusoidal signal on EOM2.
4. Conclusion
We present a new approach for the magnification of small optical signals by impressing them onto an SMOS undergoing self-phase modulation. The all-optical magnification process allows for magnification by a factor $(2m+1)$ for the $m^{th}$ order SPM-generated sideband of small optical signals. Magnification of signals with a modulation depth between 50 and 180 mV have been measured to undergo a contrast magnification of respectively 3 and 5 for the first and second order sidebands, in accordance with theoretical predictions. This all-optical signal magnification technique will allow for enhanced point sensing capabilities by bringing small signals of interest within the range of detection of conventional photodetectors.
Funding
Natural Sciences and Engineering Research Council of Canada (7RGPIN-2015-06071); Canada Research Chairs (950-231352).
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