# Integrated Fabry–Perot Cavities: A Quantum Leap in Technology

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## Definition

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## 1. The Advent of Integrated Fabry–Perot Cavities

_{M}), have been at the heart of recent advancements in optical telecommunication. They play a pivotal role in various signal processing functions [4], including channel-drop filtering, on–off switching and light modulation. Moreover, the exaltation of optical non-linearity and spontaneous emission inhibition or exaltation is primarily governed by the Q/V

_{M}ratio. Thus, the quest for ultra-small high-Q cavities [5] has been critical in optical telecommunications [6] and in quantum optics [7]. Several strategies have been proposed to obtain high-Q cavities, like photonic crystals [8], micro-rings, micro-disks [9,10], micro-toroids [11], micro-spheres [12], meta-materials [13], plasmonics [14] and other interesting schemes [15]. However, each of these strategies either produces very large Q factors in excess of 10

^{6}, but with rather large volumes, or, for the case of plasmonic resonators, a non-CMOS-compatible fabrication process. This entry will focus on arguably the most promising approach, Fabry–Perot cavities. It is believed that a Q factor as large as 10

^{8}can be achieved with cavities offering mode volumes close to the theoretical limit of V

_{M}$={(\lambda /2n)}^{3}$, where $\lambda $ is the resonant wavelength and n is the waveguide’s refractive index.

## 2. The Challenges with Current Cavities

## 3. The Design of Integrated Fabry–Perot Cavities for Classical Light

_{g}is the group velocity of the cavity mode and ${\lambda}_{0}$ the resonant wavelength.

_{m}, the transmission T

_{m}and the scattering losses L

_{m}. R

_{m}is determined by the length of the periodic mirror and by the loss management due to the tapering process. This is due to energy conservation, which imposes that ${R}_{m}+{T}_{m}+{L}_{m}=1$ or, in other words, all the energy is accounted for by the reflection, transmission and loss coefficients. Considering an infinite mirror allows us to derive an upper limit on the reflection: ${R}_{m}=1-{L}_{m}$. As for Bragg mirrors, ${R}_{m}$ is a function of the period number and the refractive index contrast. Therefore, the real upper limit is dictated by the losses. In order to obtain a very large Q factor, a simple condition, albeit difficult to achieve in integrated optics, is needed, namely ${R}_{m}>>{T}_{m}>>{L}_{m}$. Note that the losses L

_{m}are mainly due to the scattering of light into the external radiation continuum of modes. Indeed, when light propagates in a waveguide mode, with a certain mode profile, the transition into another is characterized by a coupling coefficient and a loss, like the transition in microwaves between two regions of different impedance. The coupling and the loss coefficients are expressed as a function of an overlap integral ($\eta $) between the mode profiles of the two regions, as described in [24]. When the two modes are very much alike, the overlap is close to one and therefore the coupling is as well, whereas the losses can be obtained with a good approximation as L

_{m}= 1−${\eta}^{2}$. The Bloch modes depend mainly on the geometry of the index modulation profile, and it can be shown that small holes at an appropriate period $\mathrm{\Lambda}/{\lambda}_{0}$, for a specific wavelength ${\lambda}_{0}$, exhibit lower losses. However, R

_{m}drops drastically as well, implying a much longer mirror and larger cavity volume. The main idea of the tapering process is to bring gradually the mode profile of the cavity mode close to the one of the mirror. This transition cannot be obtained adiabatically; otherwise, L

_{p}grows exponentially [25] and can be computed numerically [23] (even a simple linear transition is beneficial). Although the adaptation is done at a single wavelength, it is interesting to note that it is quite broadband as the roll-off is not excessively steep. This explains the relative robustness of the technique against fabrication variations. It is important to note that the transmission coefficient T

_{m}could be made arbitrarily small by increasing the number of periods in the mirror. However, due to the loss channel available, the resonant behavior of the cavity also boosts the losses at resonance—see [6]—and any energy that is not transmitted is lost by the coupling to the radiation continuum. As a result, the maximum transmission at resonance is obtained and is equal to ${T}_{MAX}=\frac{1}{{(1+{L}_{m}/{T}_{m})}^{2}}$. This simple relation indicates that the peak intensity at transmission vanishes to zero with T

_{m}and therefore limits the practical value of the transmission coefficient. It is possible to derive the general expression for the Q factor based on the parameters defined previously:

_{g}, as in [26]; to augment the mirror reflectivity using the method described previously; or finally to increment the cavity length. Thanks to this mode engineering, for the first time, in 2006, integrated Fabry–Perot cavities demonstrated impressive performance [17]. Experimental results have shown that these cavities can achieve a Q factor as high as 58,000, with a modal volume V

_{m}of approximately 0.6($\lambda $/n)

^{3}. This led to a Q/V ratio as high as 1.0 10

^{5}($\lambda $/n)

^{3}, representing a significant milestone in the field and proving the viability of the approach.

_{M}. On the far right, another light vertical band appears, indicating another bandgap or reflection band due to the periodic nature of the structure. One can also notice some dark vertical bands that correspond to a range of wavelengths where the cavity is “transparent” and almost no reflection occurs.

## 4. Quantum Light in Cavity

_{M}is a direct reflection of the interaction between light and matter. As such, IFPCs have emerged as one of the most crucial tools for quantum information technologies. They can effectively isolate an atom–photon system from the external environment and significantly boost weak interactions between light and matter. The non-linear interactions, either through high-order susceptibility (${\chi}^{\left(2\right)},{\chi}^{\left(3\right)}$) or atomic interaction with the electronic dipole (quantum dots), are the key components to obtain quantum light. Hence, this capacity offers the perfect basis for cavity quantum electrodynamics (CQED), which has been confined to very complex cryogenic systems. CQED is expected to be a key facilitator of the future “quantum internet”. This technological breakthrough would be based on devices that mediate quantum entanglement between photons and atomic emitters. For instance, across a “quantum” network, specific nodes are defined, where quantum states (i.e., qubits) are stored and processed. Other nodes are enabled to broadcast through optical connections from entangled states to the whole network. It has been proposed that these networks might leverage arrays of tunable optical cavities, on a single chip or a series of interconnected chips, where atoms can be introduced in specific positions and trapped in order to create qubit memories [28]. However, the technological challenge is not only to create the cavity but to include an emitter inside the cavity [29]. It is not only the interplay between the atom and cavity that is at the center of quantum light generation, but also the ability to place the emitter at a strategic position, namely the maxima of the electric field, which is crucial. Some demonstrations [30,31] have shown the potential for this, but it remains elusive.

## 5. Fundamental Principles of CQED

_{M}is the fundamental cavity mode volume. In Figure 2, a red cross indicates the best location for a single quantum dot, as in [33]. This coupling rate is the main parameter that governs the system’s behavior and two main regimes can be identified. Again, it can be shown that this coupling is directly linked to the ratio Q/V

_{M}.

#### 5.1. Strong Coupling Regime

#### 5.2. Weak Coupling Regime

_{0}= 1/C, is termed the critical atom number and can be interpreted as the number of atoms required to significantly affect the cavity field. Therefore, large cooperativity indicates large coupling between the cavity mode and the atomic system, even with the addition of losses. In fact, the emitter placement greatly affects C and constitutes, in practice, the main parameter limiting it.

#### 5.3. The Purcell Regime

#### 5.4. Heralded Single Photon Emitters

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

HPC | High-Performance Computing |

IFPC | Integrated Fabry–Perot Cavity |

Q | Quality Factor |

V_{M} | Modal Volume |

CMOS | Complementary Metal Oxide Semiconductor |

NLAM | Non-Linear Atomic Memory |

CQED | Cavity Quantum Electrodynamics |

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**Figure 1.**Diagram of an integrated Fabry–Perot cavity. (

**A**) General structure and definition of the physical cavity length. (

**B**) Definition of the different components of a mirror described in terms of cells and of period $\mathrm{\Lambda}$. (

**C**) Description of possible cell types that can be found in the literature.

**Figure 3.**Reflection calculation of a laminar system composed of high and low index layers as a function of the dimensionless factor h/$\lambda $ and the normalized cavity length h/h

_{0}. Here, the nominal cavity resonance ${\lambda}_{0}$ is set at a wavelength of 1550 nm, corresponding to a cavity length h

_{0}of 383 nm.

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Velha, P.
Integrated Fabry–Perot Cavities: A Quantum Leap in Technology. *Encyclopedia* **2024**, *4*, 622-629.
https://doi.org/10.3390/encyclopedia4020039

**AMA Style**

Velha P.
Integrated Fabry–Perot Cavities: A Quantum Leap in Technology. *Encyclopedia*. 2024; 4(2):622-629.
https://doi.org/10.3390/encyclopedia4020039

**Chicago/Turabian Style**

Velha, Philippe.
2024. "Integrated Fabry–Perot Cavities: A Quantum Leap in Technology" *Encyclopedia* 4, no. 2: 622-629.
https://doi.org/10.3390/encyclopedia4020039