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Physical Review A

Thermal phase noise in Fabry-Pérot resonators and fiber Bragg gratings

George Skolianos, He Wen, Michel J. F. Digonnet

Published Mar 11, 2014• 14 citations
Publisher

Abstract

We report a rigorous derivation of the power spectral density (PSD) spectrum of the phase noise in the transmitted and reflected signals of a Fabry-P\'erot (FP) interferometer. In transmission off and on resonance, the PSD at low frequencies is found to be simply equal to the PSD of the single-pass phase noise (the phase noise acquired by a signal that has traveled once through the FP interferometer) times ${({n}_{g}/n)}^{2}$, where ${n}_{g}$ is the group index and $n$ the phase index of the light inside the FP interferometer. Equivalently, the phase noise (in radians) is equal to the single-pass phase-noise times ${n}_{g}/n$. This result states that the phase noise is proportional to the number of passes light makes through the FP interferometer, as expected physically. This simple expression provides a quick means of calculating the phase noise in a linear Fabry-P\'erot interferometer of any length in the range of frequencies of interest to the most applications. At high frequencies, we show that the PSD spectrum can also be expressed in a simple closed-form expression on resonance, but that off resonance it must be calculated numerically. Numerical simulations show that the PSD oscillates between 3 dB below the low-frequency PSD and 3 dB below the single-pass PSD. Equivalent expressions are derived for the phase noise in the signal reflected by a Fabry-P\'erot interferometer, which is found to have the same magnitude at low frequencies, but to be typically significantly lower at high frequencies.

DOI
10.1103/physreva.89.033818

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Reply to “Comment on ‘Thermal phase noise in Fabry-Pérot resonators and fiber Bragg gratings’ ”

George Skolianos, Michel J. F. Digonnet

The previous Comment clarifies a few key points that were unclear in an earlier paper by its authors, in particular their definitions of phase and of phase noise. We now agree that this earlier paper was modeling the resonance-frequency fluctuations of a Fabry-Perot resonator, whereas our article [Phys. Rev. A 89, 033818 (2014)] was modeling the noise in the phase of the resonator's output field. In light of the Comment, the two pictures are indeed consistent. The resonance-frequency fluctuations are independent of group index, whereas the phase noise in the output field is proportional to group index. This important dependence was pointed out in our paper [Phys. Rev. A 89, 033818 (2014)]. From a more physical standpoint, we argue that phase noise can be described without resorting to the concept of signal-to-noise ratio and that, although the phase noise is of course in part a property of the fiber itself, its group-index dependence is introduced solely by the resonator.

Physical Review A•Jul 17, 2015•0 citations

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We calculate the intensity fluctuations of the light transmitted by a Fabry-P\'erot interferometer when a broadband laser is incident upon it. The laser field is assumed to have a Lorentzian spectrum and to be broadened by random amplitude fluctuations (real or complex) or random frequency fluctuations. In each case we obtain closed-form expressions for the intensity autocorrelation function, from which we can obtain the magnitude of the intensity fluctuations, as well as the power spectrum. These results indicate a strong dependence of the transmitted intensity fluctuations on the higher-order statistical properties of the incident field. Agreement with similar results concerning atomic fluorescence fluctuations is very good.

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The exact expressions of finesse and full width at half maximum (FWHM) of the Airy distribution of a Fabry-P\'erot (FP) resonator are derived, which solves the breakdown problem at ultralow reflectivity in traditional formulas. We demonstrate that, when the cavity-mirror reflectivity $R\ensuremath{\rightarrow}0$, the FWHM approaches half of the free spectral range rather than infinity; moreover, the cavity finesse $\mathcal{F}$ approaches 2 rather than zero. These expressions are useful for the development of bad-cavity lasers, such as the four-level active optical clock based on the strong atomic transition of cesium. Also, the exact expressions of the FWHM and finesse of the reflection distribution composed of reflected mode profiles are derived, which separately intersect with that of the Airy one at $R=0$. In addition, the characteristics of an antiresonant cavity, the frequency of which is exactly at the center of two adjacent resonant cavity modes, are analyzed. It is demonstrated that at the resonant and antiresonant frequencies the enhancement and inhibition of intracavity light intensity caused by the FP cavity are symmetrical in the logarithmic view. Furthermore, we provide a universal expression of the cavity-enhancement factor rather than the classical representation of ``$\text{2}\mathcal{F}/\phantom{\text{2}\mathcal{F}\ensuremath{\pi}}\phantom{\rule{0.0pt}{0ex}}\ensuremath{\pi}$,'' which is inadequate for the high-loss cavity because the cavity-enhancement factor should naturally reach 1 when $R\ensuremath{\rightarrow}0$. Finally, we extend the application of the antiresonant cavity to an inhibited laser, the frequency of which has a stronger suppression effect on the cavity-length thermal noise than the traditional resonant laser.

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Coherence and intensity statistics in phase-conjugate resonators

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The spectra, coherence properties, and intensity fluctuations of waves generated by reflections of a noisy probe field from a phase-conjugate Fabry-Perot resonator are discussed. The emphasis of the analysis is on the spectral response of the resonator. The results obtained are in fill agreement with those derived using specific noise models in the time domain. Advantages of the frequency-domain analyses include full tractability of general probe-wave noise models and fast convergence of numerical computations. The reflected-wave coherence properties reveal an interesting periodic structure due to multiple intracavity reflections. The intensity fluctuation levels, which depend on the statistical models, are calculated explicitly for pure Markovian phase noise and Gaussian amplitude noise of the probe. Combinations of these two, yielding simple laser models, and other extensions of noise effects in phase-conjugate resonators are briefly discussed.>

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Quantum Fluctuations in the Small Fabry–Perot Interferometer

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Spectra of the small Fabry–Perot interferometer (FPI) of the size of the order of the wavelength, with the main mode excited by a quantum field from a nano–LED or a laser, are investigated. The input field is detuned from the FPI mode with only a few photons. We formulate the convenient model for the FPI interacting with a quantum field, and provide novel explicit expressions for the field and the photon number fluctuation spectra inside and outside the FPI, with clearly identified contributions of the quantum and the classical noise. As a result, we found the spectra structures are quite different for the field, the photon number fluctuations inside the FPI, for the transmitted and the reflected fields and note asymmetries in spectra. The quantum noise is colored (or white) inside (or outside) the FPI, which explains differences in spectra. As another novel result, we calculate the second-order time auto–correlation functions for the FPI field; they oscillate and are negative under certain conditions. Results will help the study, design, manufacture, and use of the small elements of quantum optical integrated circuits, such as delay lines or optical transistors.

arXiv:2212.13430•Symmetry•Jan 27, 2023•3 citations
We report a rigorous derivation of the power spectral density (PSD) spectrum of the phase noise in the transmitted and reflected signals of a Fabry-P'erot (FP) interferometer. In transmission off and on resonance, the PSD at low frequencies is found to be simply equal to the PSD of the single-pass phase noise (the phase noise acquired by a signal that has traveled once through the FP interferometer) times (ng/n)2{({n}_{g}/n)}^{2}(ng​/n)2, where ng{n}_{g}ng​ is the group index and nnn the phase index of the light inside the FP interferometer. Equivalently, the phase noise (in radians) is equal to the single-pass phase-noise times ng/n{n}_{g}/nng​/n. This result states that the phase noise is proportional to the number of passes light makes through the FP interferometer, as expected physically. This simple expression provides a quick means of calculating the phase noise in a linear Fabry-P'erot interferometer of any length in the range of frequencies of interest to the most applications. At high frequencies, we show that the PSD spectrum can also be expressed in a simple closed-form expression on resonance, but that off resonance it must be calculated numerically. Numerical simulations show that the PSD oscillates between 3 dB below the low-frequency PSD and 3 dB below the single-pass PSD. Equivalent expressions are derived for the phase noise in the signal reflected by a Fabry-P'erot interferometer, which is found to have the same magnitude at low frequencies, but to be typically significantly lower at high frequencies.
The previous Comment clarifies a few key points that were unclear in an earlier paper by its authors, in particular their definitions of phase and of phase noise. We now agree that this earlier paper was modeling the resonance-frequency fluctuations of a Fabry-Perot resonator, whereas our article [Phys. Rev. A 89, 033818 (2014)] was modeling the noise in the phase of the resonator's output field. In light of the Comment, the two pictures are indeed consistent. The resonance-frequency fluctuations are independent of group index, whereas the phase noise in the output field is proportional to group index. This important dependence was pointed out in our paper [Phys. Rev. A 89, 033818 (2014)]. From a more physical standpoint, we argue that phase noise can be described without resorting to the concept of signal-to-noise ratio and that, although the phase noise is of course in part a property of the fiber itself, its group-index dependence is introduced solely by the resonator.
We calculate the intensity fluctuations of the light transmitted by a Fabry-P\'erot interferometer when a broadband laser is incident upon it. The laser field is assumed to have a Lorentzian spectrum and to be broadened by random amplitude fluctuations (real or complex) or random frequency fluctuations. In each case we obtain closed-form expressions for the intensity autocorrelation function, from which we can obtain the magnitude of the intensity fluctuations, as well as the power spectrum. These results indicate a strong dependence of the transmitted intensity fluctuations on the higher-order statistical properties of the incident field. Agreement with similar results concerning atomic fluorescence fluctuations is very good.
The exact expressions of finesse and full width at half maximum (FWHM) of the Airy distribution of a Fabry-P\'erot (FP) resonator are derived, which solves the breakdown problem at ultralow reflectivity in traditional formulas. We demonstrate that, when the cavity-mirror reflectivity $R\ensuremath{\rightarrow}0$, the FWHM approaches half of the free spectral range rather than infinity; moreover, the cavity finesse F\mathcal{F}F approaches 2 rather than zero. These expressions are useful for the development of bad-cavity lasers, such as the four-level active optical clock based on the strong atomic transition of cesium. Also, the exact expressions of the FWHM and finesse of the reflection distribution composed of reflected mode profiles are derived, which separately intersect with that of the Airy one at R=0R=0R=0. In addition, the characteristics of an antiresonant cavity, the frequency of which is exactly at the center of two adjacent resonant cavity modes, are analyzed. It is demonstrated that at the resonant and antiresonant frequencies the enhancement and inhibition of intracavity light intensity caused by the FP cavity are symmetrical in the logarithmic view. Furthermore, we provide a universal expression of the cavity-enhancement factor rather than the classical representation of ``$\text{2}\mathcal{F}/\phantom{\text{2}\mathcal{F}\ensuremath{\pi}}\phantom{\rule{0.0pt}{0ex}}\ensuremath{\pi}$,'' which is inadequate for the high-loss cavity because the cavity-enhancement factor should naturally reach 1 when $R\ensuremath{\rightarrow}0$. Finally, we extend the application of the antiresonant cavity to an inhibited laser, the frequency of which has a stronger suppression effect on the cavity-length thermal noise than the traditional resonant laser.
Extraordinary group delays shorter than the transit time of light propagating at ccc through an equal distance have been experimentally demonstrated in single-Fabry-P\'erot (FP) waveguide systems and cascaded-FP structures under off-resonant conditions. These ``superluminal'' phenomena are well explained by the multiple-reflection destructive interference that reduces the intracavity stored energy when operating off resonances. Excellent agreement between theory and experiment is obtained when the dispersive effects of reflective boundaries are considered. These results provide further insight into the nature of apparent superluminality in regions of allowed propagation.
The spectra, coherence properties, and intensity fluctuations of waves generated by reflections of a noisy probe field from a phase-conjugate Fabry-Perot resonator are discussed. The emphasis of the analysis is on the spectral response of the resonator. The results obtained are in fill agreement with those derived using specific noise models in the time domain. Advantages of the frequency-domain analyses include full tractability of general probe-wave noise models and fast convergence of numerical computations. The reflected-wave coherence properties reveal an interesting periodic structure due to multiple intracavity reflections. The intensity fluctuation levels, which depend on the statistical models, are calculated explicitly for pure Markovian phase noise and Gaussian amplitude noise of the probe. Combinations of these two, yielding simple laser models, and other extensions of noise effects in phase-conjugate resonators are briefly discussed.>
Spectra of the small Fabry–Perot interferometer (FPI) of the size of the order of the wavelength, with the main mode excited by a quantum field from a nano–LED or a laser, are investigated. The input field is detuned from the FPI mode with only a few photons. We formulate the convenient model for the FPI interacting with a quantum field, and provide novel explicit expressions for the field and the photon number fluctuation spectra inside and outside the FPI, with clearly identified contributions of the quantum and the classical noise. As a result, we found the spectra structures are quite different for the field, the photon number fluctuations inside the FPI, for the transmitted and the reflected fields and note asymmetries in spectra. The quantum noise is colored (or white) inside (or outside) the FPI, which explains differences in spectra. As another novel result, we calculate the second-order time auto–correlation functions for the FPI field; they oscillate and are negative under certain conditions. Results will help the study, design, manufacture, and use of the small elements of quantum optical integrated circuits, such as delay lines or optical transistors.