Abstract

We demonstrate the enhancement of the resolution of a fiber optical sensor using all-optical signal processing. By sweeping the frequency of a tunable laser across a fiber Bragg grating, a signal corresponding to the reflection spectrum of the FBG is generated. If another laser with fixed power and frequency is launched into a highly nonlinear fiber along with the FBG-shaped signal, the Kerr effect gives rise to a number of frequency sidebands, where the power in each of the sidebands is proportional an integer exponent of the signal and pump powers. By filtering out particular sidebands, this potentiation effect reduces the width of the FBG-shaped signal, making shifts in its central wavelength easier to distinguish. We report a maximum resolution enhancement factor of 3.35 obtained by extracting the n = −4 order sideband, and apply resolution enhancement to improve the resolution of an FBG based temperature sensor. The method described in this paper can be applied to existing fiber based sensors and optical systems to enhance their resolution.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In high speed telecommunication, optical fibers have become ubiquitous due to their bandwidth and immunity to electromagnetic interference. Light guided by total internal reflection in a silica core can travel dozens of kilometers without significant loss of intensity. Moreover, novel techniques have been developed to allow optical fibers to measure temperature, strain, vibration and other critical information about their surrounding environment [1]. A simple example involves shining broadband light onto a Fiber Bragg Grating (FBG) and measuring the reflected spectrum. Because a change in temperature will change the spatial period of the grating, thereby altering the resonantly reflected wavelength, a careful calibration against a known temperature change yields a reliable thermometer. The resolution of such a measuring scheme depends on the ability to determine small changes in the central reflected wavelength. Ideally, an infinitely narrow reflection spectrum would allow detection of minuscule temperature changes, because smaller changes in the central wavelength of the grating can be distinguished. State of the art, commercially available $\pi$-shifted gratings have spectral widths of a few picometers, allowing high resolution measurements to be conducted [2]. Further reduction in spectral width requires either new fabrication techniques or novel methods for processing the signals from existing gratings.

The Kerr effect enables a number of techniques for manipulating signals by all-optical means. One such technique utilizes the power-dependent nature of the Kerr effect to improve the extinction ratio of signals and to magnify small signal fluctuations [3,4]. Another technique to sensitivity enhancement using the Kerr effect exploits the fact that a change in the frequency difference between two signals entering the Kerr medium causes the generated sidebands to shift by the same frequency change multiplied by an order-dependent integer [5].

In this paper, we present a novel technique for enhancing the resolution of FBG-based temperature sensors via all-optical signal processing based on the Kerr effect. First, theory shows that propagation of light composed of two different frequencies, a “pump” and a “signal”, in a Kerr medium leads to the generation of sidebands. The power of the sidebands is proportional to integer exponents of the input power and the implications of this potentiation effect to the measurement of optical signals are discussed. The potentiation dependence of the sideband power on the input signal is demonstrated experimentally by extracting sidebands up to the $n=-4$ order and applied to enhance the resolution of an FBG based temperature sensor. Subsequently, further applications of the resolution enhancement effect to existing optical sensing systems are discussed.

2. Theory

The following description of the generation of sidebads in a Kerr medium is based on the approach used in [6]. Launching laser light with two different angular fequencies, a signal $\omega _{s}$ and a pump $\omega _{p}$ with $\omega _{s}<\omega _{p}$, into a Kerr medium consisting of a Highly-Nonlinear-Fiber (HNLF), the total electric field amplitude is given by $A_{in}=\left [\sqrt {P_{s}}\exp \left (-0.5i\omega _{d}t\right )+\sqrt {P_{p}}\exp \left (0.5i\omega _{d}t\right )\right ]$, where $P_{s/p}$ is the power of the signal/pump fields, $\omega _{d}=\omega _{p}-\omega _{s}$, and the power of this input field is $|A_{in}|^{2}=\left [P_{s}+P_{p}+2\sqrt {P_{s}P_{p}}\cos \left (\omega _{d}t\right )\right ]$. Neglecting dispersion, loss and polarization effects, this incident field evolves according to the Non-linear Schrödinger Equation, $dA/dz=i\gamma |A|^{2}$ [7]. Solving this differential equation, the field at the output of the fiber is $A_{out}=A_{in}\exp \left [i\gamma L\left (P_{s}+P_{p}\right )\right ]\exp \left [i\gamma L\cdot 2\sqrt {P_{s}P_{p}}\cos \left (\omega _{d}t\right )\right ]$. The exponential factor $\exp \left [i\gamma L\left (P_{s}+P_{p}\right )\right ]$ is neglected as it does not affect the power of the output field. The Jacobi-Anger expansion, $\exp \left [iM\cos \left (\Omega t\right )\right ]=\sum _{n=-\infty }^{\infty }i^{n}J_{n}(M)\exp \left (in\Omega t\right )$ [8], is applied to the factor $\exp \left [i\gamma L\cdot 2\sqrt {P_{s}P_{p}}\cos \left (\omega _{d}t\right )\right ]$, allowing the output field to be expressed as the sum of a number of frequency sidebands spaced $\omega _{d}$ apart:

$$\begin{aligned} A_{out}= & \sum_{n={-}\infty}^{\infty}i^{n}e^{i\omega_{d}\left(n+\frac{1}{2}\right)t}\left[i\cdot J_{n+1}\left(2\gamma L\sqrt{P_{s}P_{p}}\right)\sqrt{P_{s}}+J_{n}\left(2\gamma L\sqrt{P_{s}P_{p}}\right)\sqrt{P_{p}}\right],\\ \end{aligned}$$
where $J_{n}$ is the $n^{\textrm {th}}$ order Bessel function of the first kind. The sideband $n=0$ corresponds to $\omega _{p}$, while $n=-1$ corresponds to $\omega _{s}$. The power of the $n^{\textrm {th}}$ order sideband is $|A_{n}|^{2}=J_{|n+1|}^{2}\left (2\gamma L\sqrt {P_{s}P_{p}}\right )P_{s}+J_{|n\textrm {|}}^{2}\left (2\gamma L\sqrt {P_{s}P_{p}}\right )P_{p}$, and introducing the normalization $x=\gamma LP_{s}$, $y=\gamma LP_{p}$, $z_{n}=\gamma L|A_{n}|^{2}$ yields $z_{n}=xJ_{|n+1|}^{2}\left (2\sqrt {xy}\right )+yJ_{|n|}^{2}\left (2\sqrt {xy}\right )$. The Bessel function $J_{n}(\textrm{M})$ can be approximated by $1/n!\left (M/2\right )^{n}$ under the condition $0<M<\sqrt {1+n}$ [9]. Using this approximation, the power in the $n^{\textrm {th}}$ order sideband becomes
$$\begin{aligned} z_{n}\approx & x^{|n+1|+1}y^{|n+1|}\frac{1}{((|n+1|)!)^{2}}+x^{|n|}y^{|n|+1}\frac{1}{(|n|!)^{2}}.\\ \end{aligned}$$
Eq. (2) shows that the normalized output power is proportional to the normalized input power raised to an integer exponent. As an example, filtering out the $n=-2$ sideband yields $z_{-2}\approx x^{2}\left (y+ y^{3}/4\right)$, so an input signal, $x$, leads to an output signal proportional to $x^{2}$. This potentiation effect is utilized to enhance the resolution of fiber optical sensors. Consider a generic sub/super-Gaussian spectrum with Full Width Half Maximum (FWHM)$=2\Lambda$. The input signal, $x_{in}(\lambda )\propto \exp \left (-\ln (2)\cdot \lambda ^{m}/\Lambda ^{m}\right )$, yields the output signal, $x_{out}\propto \exp \left (-\ln (2)\cdot \lambda ^{m}/\left (\Lambda _{n}\right )^{m}\right )$, where $2\Lambda _{n}=2\Lambda /n^{1/m}<2\Lambda$ is the FWHM of the output signal. If $x_{in}(\lambda )$ is generated by sweeping the wavelength of a laser at a constant rate across an FBG, time dependence can be converted to wavelength dependence and the output signal will be a narrower version of the FBG spectrum. The theoretical decrease in width for various sub/super-Gaussian signals is shown in Fig. 1(a), while Fig. 1(b-d) illustrates that signals with sharp peaks benefit more from resolution enhancement than signals with flat tops.

 figure: Fig. 1.

Fig. 1. (a) Decreasing FWHM for different sub/super-Gaussian spectra as the sideband order increases. (b-d) Examples of narrowing for different spectrum types and sideband orders normalized to the peak value for $n=1,2,3$. Such signals can be generated by sweeping the wavelength of a tunable laser across an FBG and converting time dependence to wavelength dependence by using the sweeprate of the laser.

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3. Experimental setup

An illustration of the experimental setup utilized for resolution enhancement with the Kerr effect is presented in Fig. 2(a). The signal is generated by a tunable laser (Agilent 81940A) that is swept at a rate of $50$ nm/s across $60$ pm from $1550.08$ nm to $1550.14$ nm. Using a 1:99 coupler, 1 $\%$ of this light is sent through an auxillary filter (TeraXion C095381 $30$ pm PSG $1550.12$ nm) that is used as a reference and passed directly to a photodiode (Thorlabs PDB130C-AC) connected to an oscilloscope (Agilent infiniium DSO81204B $12$ GHz) sampling at $2$ GSa/s. The arrival times of all other signals are measured relative to the reference. Any change in arrival time is converted into a wavelength shift by using the sweeprate of the tunable laser. The remaining 99 $\%$ of the light from the tunable laser is reflected off an FBG (TeraXion custom $24$ pm grating) located inside an oven (Yamato DX300). By changing the temperature inside the oven, the central wavelength of the FBG can be changed. When the wavelength of the light from the tunable laser is swept across the reflection spectrum of the FBG, a signal referred to as the “raw” signal is obtained because the reflected power as a function of time is identical to the reflection spectrum of the FBG. The raw signal is visualized in Fig. 2(b). Using another 1:99 coupler, $1$ % of the raw signal is extracted and passed to a photodiode connected to the oscilloscope. The remaining $99$ % of the raw signal is combined with high power pump pulses and sent into the Kerr medium (Draka Comteq 418SG 04611A).

 figure: Fig. 2.

Fig. 2. (a) Experimental setup. The magenta fiber contains the reference signal, while the blue fiber contains the raw signal from the FBG. The green fiber contains the resolution enhanced signal obtained by extracting the $n^{\textrm {th}}$ order sideband. (b-e) The spectrum and signal at successive stages of the experiment.

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To generate the high power pump pulses, a Distributed Feedback (DFB) laser (Lucent M-E2566H) centered at $1549.60$ nm is driven by a function generator (Agilent 33250A) using $10$ ns pulses with a $100$ ns period. The experiment can be conducted with a CW pump laser, but pulses are utilized to achieve higher peak powers and supress Stimulated Brillouin Scattering (SBS). The generated laser pulses are amplified using a conventional Erbium Doped Fiber Amplifier (EDFA) (GN Nettest fiberamp BT 17) followed by a High Power EDFA (Amonics AEDFA-33-B-FA). The combination of the high power pump pulses with the signal is visualized in Fig. 2(c). The Kerr effect giving rise to sidebands is illustrated in Fig. 2(d). A measurement of the spectrum immediately after the Kerr medium is presented in Fig. 3 showing the signal, the pump and the sidebands generated by the Kerr effect.

 figure: Fig. 3.

Fig. 3. Spectrum after the Kerr medium with sideband orders labelled. The $n=0$ peak corresponds to the high power pump pulses, while the $n=-1$ peak is the frequency of the tunable laser that sweeps across the FBG to generate the signal. The sidebands $n=-2,-3,-4,\ldots$ can be extracted to obtain outputs that are proportional to integer powers of the input signal.

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Individual sidebands are extracted using two filters (AlnairLabs WTF-200 $100$ pm, and JDS Uniphase TB25MI+1FP Tunable $215$ pm Etalon Filter) to ensure that the signal in the sidebands is extracted without the high power pump pulses leaking through and distorting the measurement. At this stage, the extracted sideband consists of a train of pulses, whose envelope is proportional to an integer power of the raw signal as visualized in Fig. 2(e). The signal is amplified by an EDFA (Amonics AEDFA-PA-25-B-FA), passed through a bandpass filter (JDS FITEL TB45B+1SC $500$ pm Tunable Filter) to remove ASE noise, and detected by a photodiode connected to the oscilloscope. From the trace detected by the oscilloscope, the peaks of the pulse train can be digitally extracted to obtain the envelope, which is referred to as the “sideband” signal. The measured traces corresponding to the reference, the raw signal, and the sideband signal are presented in Fig. 4(a). Examples of three sideband signals for $n=-2,-3,-4$ are presented in Fig. 4(b-d). The FWHM of the raw signal is $8.81$ pm, while the FWHM of the sideband signals are $4.78$ pm, $3.91$ pm and $2.63$ pm respectively. This corresponds to reductions in width by factors of $1.84$, $2.25$ and $3.35$ respectively in agreement with the theory in Sec. 2.

 figure: Fig. 4.

Fig. 4. (a) Example of the signal collected from the auxillary filter used as a reference (magenta), the raw signal (blue) and the sideband signal (green). (b-d) Raw signals (blue) and sideband signals (green) for different sideband orders. The raw signal raised to the appropriate integer power (light blue) is included for comparison with the sideband signal. Traces are horizontally offset for clarity.

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The oven is heated to $40$ °C, turned off and allowed to cool gradually. The internal temperature of the oven is measured with a thermocouple thermometer (TES 1310 TYPE-K). The raw- and sideband-traces produced by the orders $n=-2$ and $n=-4$ are measured for temperatures between $32$ °C and $40$ °C. The changing arrival time relative to the reference trace is determined and converted into a wavelength shift using the sweeprate of the tunable laser. The result is shown in Fig. 5. The Taylor Criterion states that two spectral peaks can be distinguished if they are more than one FWHM apart [10]. Applying the Taylor Criterion and using the fitted slopes from Fig. 5, the smallest discernable temperature difference is $8.81$ pm $\cdot$ $0.092$ °C/pm$=0.81$°C for the raw signal. For the $n=-2$ and $n=-4$ sideband traces, the temperature resolutions according to the Taylor Criterion are $0.45$ °C and $0.24$ °C respectively.

 figure: Fig. 5.

Fig. 5. (a) Temperature calibration done with the $n=-2$ order sideband. (b) Temperature calibration done with the $n=-4$ order sideband. Data has been horizontally offset for clarity.

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4. Discussion

A number of simplifying assumptions are used in the derivation in Sec. 2. It is assumed that both the signal and the pump pulses can be considered quasi-CW. For the quasi-CW assumption to be applicable, the sinusoidal modulation created by the beat of the distinct pump and signal frequencies must have a period significantly shorter than the duration of the signal or pulse. In other words, the spectral width of the raw signal and the pump in Fig. 3 must be smaller than the frequency spacing between them, which in the present case is $62.4$ GHz leading to a beat period of $16$ ps. Because the beat period of $16$ ps is much shorter than the pump pulse duration of $10$ ns, the quasi-CW assumption is valid. Dispersion and loss are also assumed to be negligible and polarization effects are ignored. A model that takes dispersion, loss and polarization into account will be investigated in future works.

The resolution enhancement effect is greater for higher order sidebands, but according to Eq. (2), the power in the sidebands decreases as $1/(n!)^{2}$ for higher orders. Thus, while the magnitude of the resolution enhancement does not depend on the sideband power, extraction of higher order sidebands requires high input power, making the ability of the optical components to handle such levels of power a limiting factor. Cascading multiple resolution enhancement systems could allow for further resolution enhancement without risking damage due to high power.

The resolution enhancement effect studied in this work was only applied to a single FBG, but could be generalized to distributed gratings. Also, atomic and molecular transition lines studied via Tunable Diode Laser Absorption Spectroscopy (TDLAS) could be measured more accurately if resolution enhancement were applied [11]. Another potential application is to more precisely determine the transmission spectra of Fabry-Pérot (FP) cavities. A recent work determined that if the ends of an FP cavity have reflectivities $R_{1}$ and $R_{2}$, where their geometric mean, $\sqrt {R_{1}R_{2}}$, is greater than or equal to $17.2$ %, individual Lorentzian lines of the cavity’s transmission spectrum can be distingished subject to the Taylor Criterion [12]. For geometric means of the reflectivities below $17.2$ %, measuring the transmission as a function of frequency yields a spectrum that never drops below $0.5$, violating the Taylor Criterion. Applying the resolution enhancement technique, the smallest geometric mean of the reflectivities that produces a transmission spectrum obeying the Taylor Criterion will be $\left [\sqrt {R_{1}R_{2}}\right ]_{n}=(2^{1/2n}-1)^{2}/(2^{1/n}-1)$ yielding values of $8.64$ %, $5.77$ % and $4.35$ % for the sidebands $n=-2,-3,$ and $-4$, respectively.

Another possibility is enhancement of the resolution of diffraction limited imaging systems used in astronomy and condensed matter physics [1315]. If a light-emitting point is imaged by an ideal lens onto a screen in the focal plane of the lens, an Airy pattern consisting of a central, bright spot surrounded by alternating dark and bright rings will appear on the screen as determined by the principles of Fourier optics [16]. The width of the smeared out central spot of the Airy pattern on the screen limits the ability to detect changes in the position of the light emitting point. If one images the point onto a collection of optical fibers in the focal plane and subsequently applies the resolution enhancement technique, it would be possible to reduce the FWHM of the detected spot by factors $1.39$, $1.69$ and $1.96$ for sidebands $n=-2,-3,$ and $-4$ respectively.

5. Conclusion

Resolution enhancement based on the Kerr effect is demonstrated for sideband orders up to $n=-4$. The effect allows the measured width of the reflection spectrum of an FBG to be reduced by purely optical means, thereby improving the resolution of a temperature sensing system by a factor $3.35$ in accordance with theoretical predictions. The resolution enhancement effect can be applied to a wide range of existing optical sensing systems to improve their resolutions.

Funding

Natural Sciences and Engineering Research Council of Canada (RGPIN-2015-06071, STPGP 506628); Canada Research Chairs (950231352).

Disclosures

The authors declare no conflicts of interest.

References

1. W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber Optic Bragg Grating Sensors,” Fiber Opt. and Laser Sens. VII 1169, 98–107 (1990). [CrossRef]  

2. G. P. Agrawal and S. Radic, “Phase-shifted fiber bragg gratings and their application for wavelength demultiplexing,” IEEE Photonics Technol. Lett. 6(8), 995–997 (1994). [CrossRef]  

3. C. Baker, B. Vanus, M. Wuilpart, L. Chen, and X. Bao, “Enhancement of optical pulse extinction-ratio using the nonlinear kerr effect for phase-otdr,” Opt. Express 24(17), 19424–19434 (2016). [CrossRef]  

4. B. Vanus, C. Baker, L. Chen, and X. Bao, “All-optical intensity fluctuation magnification using kerr effect,” Opt. Express 28(3), 3789–3794 (2020). [CrossRef]  

5. J. Du, L. Li, X. Fan, Q. Liu, and Z. He, “Sensitivity enhancement for fiber bragg grating sensors by four wave mixing,” Photonics 2(2), 426–439 (2015). [CrossRef]  

6. A. Boskovic, S. V. Chernikov, J. R. Taylor, L. Gruner-Nielsen, and O. A. Levring, “Direct continuous-wave measurement of n2 in various types of telecommunication fiber at 1.55 μm,” Opt. Lett. 21(24), 1966–1968 (1996). [CrossRef]  

7. G. Agrawal, “Chapter 4 - self-phase modulation,” in “Nonlinear Fiber Optics (Fifth Edition),”G. Agrawal, ed. (Academic Press, Boston, 2013), Optics and Photonics, pp. 87–128, fifth edition ed.

8. “NIST Digital Library of Mathematical Functions,” http://dlmf.nist.gov/, Release 1.0.27 of 2020-06-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

9. G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library (Cambridge University Press, 1995).

10. M. Francon, Optical Interferometry: By m. Francon (Academic Press, New York, 1966).

11. D. T. Cassidy and J. Reid, “Atmospheric pressure monitoring of trace gases using tunable diode lasers,” Appl. Opt. 21(7), 1185–1190 (1982). [CrossRef]  

12. M. Pollnau and M. Eichhorn, “Spectral coherence, Part I: Passive-resonator linewidth, fundamental laser linewidth, and Schawlow-Townes approximation,” Prog. Quantum Electron. 72, 100255 (2020). [CrossRef]  

13. S. Saha, Diffraction-Limited Imaging with Large and Moderate Telescopes (Singapore ; Hackensack, NJ, 2007).

14. J. H. Denschlag, J. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. Rolston, and W. D. Phillips, “A bose-einstein condensate in an optical lattice,” J. Phys. B: At., Mol. Opt. Phys. 35(14), 3095 (2002). [CrossRef]  

15. E. Betzig, “Excitation strategies for optical lattice microscopy,” Opt. Express 13(8), 3021–3036 (2005). [CrossRef]  

16. E. Hecht, Optics (Pearson Education, Incorporated, 2017).

References

  • View by:

  1. W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber Optic Bragg Grating Sensors,” Fiber Opt. and Laser Sens. VII 1169, 98–107 (1990).
    [Crossref]
  2. G. P. Agrawal and S. Radic, “Phase-shifted fiber bragg gratings and their application for wavelength demultiplexing,” IEEE Photonics Technol. Lett. 6(8), 995–997 (1994).
    [Crossref]
  3. C. Baker, B. Vanus, M. Wuilpart, L. Chen, and X. Bao, “Enhancement of optical pulse extinction-ratio using the nonlinear kerr effect for phase-otdr,” Opt. Express 24(17), 19424–19434 (2016).
    [Crossref]
  4. B. Vanus, C. Baker, L. Chen, and X. Bao, “All-optical intensity fluctuation magnification using kerr effect,” Opt. Express 28(3), 3789–3794 (2020).
    [Crossref]
  5. J. Du, L. Li, X. Fan, Q. Liu, and Z. He, “Sensitivity enhancement for fiber bragg grating sensors by four wave mixing,” Photonics 2(2), 426–439 (2015).
    [Crossref]
  6. A. Boskovic, S. V. Chernikov, J. R. Taylor, L. Gruner-Nielsen, and O. A. Levring, “Direct continuous-wave measurement of n2 in various types of telecommunication fiber at 1.55 μm,” Opt. Lett. 21(24), 1966–1968 (1996).
    [Crossref]
  7. G. Agrawal, “Chapter 4 - self-phase modulation,” in “Nonlinear Fiber Optics (Fifth Edition),”G. Agrawal, ed. (Academic Press, Boston, 2013), Optics and Photonics, pp. 87–128, fifth edition ed.
  8. “NIST Digital Library of Mathematical Functions,” http://dlmf.nist.gov/ , Release 1.0.27 of 2020-06-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.
  9. G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library (Cambridge University Press, 1995).
  10. M. Francon, Optical Interferometry: By m. Francon (Academic Press, New York, 1966).
  11. D. T. Cassidy and J. Reid, “Atmospheric pressure monitoring of trace gases using tunable diode lasers,” Appl. Opt. 21(7), 1185–1190 (1982).
    [Crossref]
  12. M. Pollnau and M. Eichhorn, “Spectral coherence, Part I: Passive-resonator linewidth, fundamental laser linewidth, and Schawlow-Townes approximation,” Prog. Quantum Electron. 72, 100255 (2020).
    [Crossref]
  13. S. Saha, Diffraction-Limited Imaging with Large and Moderate Telescopes (Singapore ; Hackensack, NJ, 2007).
  14. J. H. Denschlag, J. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. Rolston, and W. D. Phillips, “A bose-einstein condensate in an optical lattice,” J. Phys. B: At., Mol. Opt. Phys. 35(14), 3095 (2002).
    [Crossref]
  15. E. Betzig, “Excitation strategies for optical lattice microscopy,” Opt. Express 13(8), 3021–3036 (2005).
    [Crossref]
  16. E. Hecht, Optics (Pearson Education, Incorporated, 2017).

2020 (2)

B. Vanus, C. Baker, L. Chen, and X. Bao, “All-optical intensity fluctuation magnification using kerr effect,” Opt. Express 28(3), 3789–3794 (2020).
[Crossref]

M. Pollnau and M. Eichhorn, “Spectral coherence, Part I: Passive-resonator linewidth, fundamental laser linewidth, and Schawlow-Townes approximation,” Prog. Quantum Electron. 72, 100255 (2020).
[Crossref]

2016 (1)

2015 (1)

J. Du, L. Li, X. Fan, Q. Liu, and Z. He, “Sensitivity enhancement for fiber bragg grating sensors by four wave mixing,” Photonics 2(2), 426–439 (2015).
[Crossref]

2005 (1)

2002 (1)

J. H. Denschlag, J. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. Rolston, and W. D. Phillips, “A bose-einstein condensate in an optical lattice,” J. Phys. B: At., Mol. Opt. Phys. 35(14), 3095 (2002).
[Crossref]

1996 (1)

1994 (1)

G. P. Agrawal and S. Radic, “Phase-shifted fiber bragg gratings and their application for wavelength demultiplexing,” IEEE Photonics Technol. Lett. 6(8), 995–997 (1994).
[Crossref]

1990 (1)

W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber Optic Bragg Grating Sensors,” Fiber Opt. and Laser Sens. VII 1169, 98–107 (1990).
[Crossref]

1982 (1)

Agrawal, G.

G. Agrawal, “Chapter 4 - self-phase modulation,” in “Nonlinear Fiber Optics (Fifth Edition),”G. Agrawal, ed. (Academic Press, Boston, 2013), Optics and Photonics, pp. 87–128, fifth edition ed.

Agrawal, G. P.

G. P. Agrawal and S. Radic, “Phase-shifted fiber bragg gratings and their application for wavelength demultiplexing,” IEEE Photonics Technol. Lett. 6(8), 995–997 (1994).
[Crossref]

Baker, C.

Bao, X.

Betzig, E.

Boskovic, A.

Browaeys, A.

J. H. Denschlag, J. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. Rolston, and W. D. Phillips, “A bose-einstein condensate in an optical lattice,” J. Phys. B: At., Mol. Opt. Phys. 35(14), 3095 (2002).
[Crossref]

Cassidy, D. T.

Chen, L.

Chernikov, S. V.

Cho, D.

J. H. Denschlag, J. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. Rolston, and W. D. Phillips, “A bose-einstein condensate in an optical lattice,” J. Phys. B: At., Mol. Opt. Phys. 35(14), 3095 (2002).
[Crossref]

Denschlag, J. H.

J. H. Denschlag, J. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. Rolston, and W. D. Phillips, “A bose-einstein condensate in an optical lattice,” J. Phys. B: At., Mol. Opt. Phys. 35(14), 3095 (2002).
[Crossref]

Du, J.

J. Du, L. Li, X. Fan, Q. Liu, and Z. He, “Sensitivity enhancement for fiber bragg grating sensors by four wave mixing,” Photonics 2(2), 426–439 (2015).
[Crossref]

Eichhorn, M.

M. Pollnau and M. Eichhorn, “Spectral coherence, Part I: Passive-resonator linewidth, fundamental laser linewidth, and Schawlow-Townes approximation,” Prog. Quantum Electron. 72, 100255 (2020).
[Crossref]

Fan, X.

J. Du, L. Li, X. Fan, Q. Liu, and Z. He, “Sensitivity enhancement for fiber bragg grating sensors by four wave mixing,” Photonics 2(2), 426–439 (2015).
[Crossref]

Francon, M.

M. Francon, Optical Interferometry: By m. Francon (Academic Press, New York, 1966).

Glenn, W. H.

W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber Optic Bragg Grating Sensors,” Fiber Opt. and Laser Sens. VII 1169, 98–107 (1990).
[Crossref]

Gruner-Nielsen, L.

Häffner, H.

J. H. Denschlag, J. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. Rolston, and W. D. Phillips, “A bose-einstein condensate in an optical lattice,” J. Phys. B: At., Mol. Opt. Phys. 35(14), 3095 (2002).
[Crossref]

He, Z.

J. Du, L. Li, X. Fan, Q. Liu, and Z. He, “Sensitivity enhancement for fiber bragg grating sensors by four wave mixing,” Photonics 2(2), 426–439 (2015).
[Crossref]

Hecht, E.

E. Hecht, Optics (Pearson Education, Incorporated, 2017).

Helmerson, K.

J. H. Denschlag, J. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. Rolston, and W. D. Phillips, “A bose-einstein condensate in an optical lattice,” J. Phys. B: At., Mol. Opt. Phys. 35(14), 3095 (2002).
[Crossref]

Levring, O. A.

Li, L.

J. Du, L. Li, X. Fan, Q. Liu, and Z. He, “Sensitivity enhancement for fiber bragg grating sensors by four wave mixing,” Photonics 2(2), 426–439 (2015).
[Crossref]

Liu, Q.

J. Du, L. Li, X. Fan, Q. Liu, and Z. He, “Sensitivity enhancement for fiber bragg grating sensors by four wave mixing,” Photonics 2(2), 426–439 (2015).
[Crossref]

McKenzie, C.

J. H. Denschlag, J. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. Rolston, and W. D. Phillips, “A bose-einstein condensate in an optical lattice,” J. Phys. B: At., Mol. Opt. Phys. 35(14), 3095 (2002).
[Crossref]

Meltz, G.

W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber Optic Bragg Grating Sensors,” Fiber Opt. and Laser Sens. VII 1169, 98–107 (1990).
[Crossref]

Morey, W. W.

W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber Optic Bragg Grating Sensors,” Fiber Opt. and Laser Sens. VII 1169, 98–107 (1990).
[Crossref]

Phillips, W. D.

J. H. Denschlag, J. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. Rolston, and W. D. Phillips, “A bose-einstein condensate in an optical lattice,” J. Phys. B: At., Mol. Opt. Phys. 35(14), 3095 (2002).
[Crossref]

Pollnau, M.

M. Pollnau and M. Eichhorn, “Spectral coherence, Part I: Passive-resonator linewidth, fundamental laser linewidth, and Schawlow-Townes approximation,” Prog. Quantum Electron. 72, 100255 (2020).
[Crossref]

Radic, S.

G. P. Agrawal and S. Radic, “Phase-shifted fiber bragg gratings and their application for wavelength demultiplexing,” IEEE Photonics Technol. Lett. 6(8), 995–997 (1994).
[Crossref]

Reid, J.

Rolston, S.

J. H. Denschlag, J. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. Rolston, and W. D. Phillips, “A bose-einstein condensate in an optical lattice,” J. Phys. B: At., Mol. Opt. Phys. 35(14), 3095 (2002).
[Crossref]

Saha, S.

S. Saha, Diffraction-Limited Imaging with Large and Moderate Telescopes (Singapore ; Hackensack, NJ, 2007).

Simsarian, J.

J. H. Denschlag, J. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. Rolston, and W. D. Phillips, “A bose-einstein condensate in an optical lattice,” J. Phys. B: At., Mol. Opt. Phys. 35(14), 3095 (2002).
[Crossref]

Taylor, J. R.

Vanus, B.

Watson, G.

G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library (Cambridge University Press, 1995).

Wuilpart, M.

Appl. Opt. (1)

Fiber Opt. and Laser Sens. VII (1)

W. W. Morey, G. Meltz, and W. H. Glenn, “Fiber Optic Bragg Grating Sensors,” Fiber Opt. and Laser Sens. VII 1169, 98–107 (1990).
[Crossref]

IEEE Photonics Technol. Lett. (1)

G. P. Agrawal and S. Radic, “Phase-shifted fiber bragg gratings and their application for wavelength demultiplexing,” IEEE Photonics Technol. Lett. 6(8), 995–997 (1994).
[Crossref]

J. Phys. B: At., Mol. Opt. Phys. (1)

J. H. Denschlag, J. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. Rolston, and W. D. Phillips, “A bose-einstein condensate in an optical lattice,” J. Phys. B: At., Mol. Opt. Phys. 35(14), 3095 (2002).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Photonics (1)

J. Du, L. Li, X. Fan, Q. Liu, and Z. He, “Sensitivity enhancement for fiber bragg grating sensors by four wave mixing,” Photonics 2(2), 426–439 (2015).
[Crossref]

Prog. Quantum Electron. (1)

M. Pollnau and M. Eichhorn, “Spectral coherence, Part I: Passive-resonator linewidth, fundamental laser linewidth, and Schawlow-Townes approximation,” Prog. Quantum Electron. 72, 100255 (2020).
[Crossref]

Other (6)

S. Saha, Diffraction-Limited Imaging with Large and Moderate Telescopes (Singapore ; Hackensack, NJ, 2007).

E. Hecht, Optics (Pearson Education, Incorporated, 2017).

G. Agrawal, “Chapter 4 - self-phase modulation,” in “Nonlinear Fiber Optics (Fifth Edition),”G. Agrawal, ed. (Academic Press, Boston, 2013), Optics and Photonics, pp. 87–128, fifth edition ed.

“NIST Digital Library of Mathematical Functions,” http://dlmf.nist.gov/ , Release 1.0.27 of 2020-06-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library (Cambridge University Press, 1995).

M. Francon, Optical Interferometry: By m. Francon (Academic Press, New York, 1966).

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Figures (5)

Fig. 1.
Fig. 1. (a) Decreasing FWHM for different sub/super-Gaussian spectra as the sideband order increases. (b-d) Examples of narrowing for different spectrum types and sideband orders normalized to the peak value for $n=1,2,3$. Such signals can be generated by sweeping the wavelength of a tunable laser across an FBG and converting time dependence to wavelength dependence by using the sweeprate of the laser.
Fig. 2.
Fig. 2. (a) Experimental setup. The magenta fiber contains the reference signal, while the blue fiber contains the raw signal from the FBG. The green fiber contains the resolution enhanced signal obtained by extracting the $n^{\textrm {th}}$ order sideband. (b-e) The spectrum and signal at successive stages of the experiment.
Fig. 3.
Fig. 3. Spectrum after the Kerr medium with sideband orders labelled. The $n=0$ peak corresponds to the high power pump pulses, while the $n=-1$ peak is the frequency of the tunable laser that sweeps across the FBG to generate the signal. The sidebands $n=-2,-3,-4,\ldots$ can be extracted to obtain outputs that are proportional to integer powers of the input signal.
Fig. 4.
Fig. 4. (a) Example of the signal collected from the auxillary filter used as a reference (magenta), the raw signal (blue) and the sideband signal (green). (b-d) Raw signals (blue) and sideband signals (green) for different sideband orders. The raw signal raised to the appropriate integer power (light blue) is included for comparison with the sideband signal. Traces are horizontally offset for clarity.
Fig. 5.
Fig. 5. (a) Temperature calibration done with the $n=-2$ order sideband. (b) Temperature calibration done with the $n=-4$ order sideband. Data has been horizontally offset for clarity.

Equations (2)

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A o u t = n = i n e i ω d ( n + 1 2 ) t [ i J n + 1 ( 2 γ L P s P p ) P s + J n ( 2 γ L P s P p ) P p ] ,
z n x | n + 1 | + 1 y | n + 1 | 1 ( ( | n + 1 | ) ! ) 2 + x | n | y | n | + 1 1 ( | n | ! ) 2 .

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