Open Access
Add to BibSonomy
Abstract
Quantum optical integrated circuits have heralded a paradigm shift in the realm of quantum information processing. Integrated photonics technology now empowers the creation of intricate optical circuits on single chips. While optical integrated circuits used to pose formidable challenges for numerous quantum applications, they have, in recent times, evolved to satisfy stringent requirements across a spectrum of research and industrial domains. Today, it is imperative to delve into research aimed at both crafting and preserving quantum properties within photonic substrates. Superimposed Bragg grating structures have emerged as valuable components within optical applications, poised to play pivotal roles in the development of integrated circuits. Nevertheless, these structures exhibit an inherent drawback in the form of dispersion, which can potentially compromise the preservation of quantum states. In this study, we meticulously scrutinize the physical attributes of these structures to elucidate the factors contributing to undesirable dispersion effects. We also investigate the correlation between two photons at the termination point of the structure. The superimposed Bragg grating structure under scrutiny boasts periods of both 1 and 3 micrometers, an overall length of 100 micrometers, and radiates at a wavelength of 1.55 micrometers. By subjecting photons to this medium individually or in tandem and analyzing their correlation function, we aim to pinpoint elements that effectively safeguard the quantum properties inherent in the system. This research endeavor is poised to yield valuable insights that will substantially influence the design of quantum integrated circuits, enhancing their efficacy in computational tasks and quantum information processing.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Harnessing photons for quantum information protocols presents a multitude of advantages owing to their facile transmission, uncomplicated manipulation, and versatile capacity to encode information utilizing diverse degrees of freedom. In recent times, light has gained substantial traction as the preferred medium for experimental investigations into quantum phenomena. This includes the exploration of quantum cryptography protocols [1] and the study of entangled light propagation within biological tissues [2]. Moreover, photons have proven invaluable in fundamental research endeavors, ranging from the examination of quantum foundations [3] to the scrutiny of light-matter interactions at the nanoscale [4,5]. The generation of photonic states predominantly relies on nonlinear processes, with spontaneous parametric down-conversion (SPDC) standing as a prime exemplar. In bulk optics SPDC, a robust pump beam is tightly focused within a nonlinear crystal, resulting in the creation of pairs of quantum-correlated photons, albeit with relatively low efficiency. The versatility of this technique lies in its capacity to manipulate these generated photon pairs across numerous degrees of freedom (DOFs). Researchers have undertaken extensive investigations into photon entanglement across various DOFs, including polarization, frequency, orbital angular momentum, and linear transverse momentum. Typically, the entanglement properties are explored by examining a single DOF of the photons. This is often achieved through filtering and post-selecting specific values of the other DOFs or engineering them to minimize entanglement. However, it is worth noting that it is also feasible to study the properties of photons that exhibit entanglement across multiple DOFs, as is the case with hyper-entangled photons [6].
The interplay between spatial and temporal (or spectral) degrees of freedom is a fundamental aspect of SPDC (spontaneous parametric down-conversion) processes and has remained a subject of keen interest for decades. For example, in Ref. [7], an investigation was conducted to unravel the relationship between the spatial coordinates of biphoton generation and the spectral bandwidth of the pump. The capacity to manipulate the spectro-temporal distribution is paramount for a variety of applications. Altering the spectral distribution of photons furnishes valuable insights into the electronic structure of a medium [8] and facilitates information processing techniques such as orthogonal spectral coding [9]. Numerous techniques exist to manipulate the distribution of entangled photons concerning their spectro-temporal properties, as elucidated in various research papers. In Ref. [10], a method was introduced for generating entangled photons with controllable frequency correlations. This approach involves initiating counter-propagating SPDC within a single-mode nonlinear waveguide using a pulsed pump beam. Reference [11] demonstrated that spectra can be controlled by shaping the spatial profile of the pump beam. More sophisticated methodologies encompass the deployment of a quantum light pulse in conjunction with a tailored classical field [12] or the direct adjustment of the poling period of the nonlinear crystal [13]. Photon pairs generated via SPDC hold immense significance in both quantum physics and information processing, primarily due to their precise temporal correlation [14–16]. When it comes to distributing these paired photons, optical fibers emerge as the ideal choice owing to their low transmission loss. However, the dispersion within the fiber leads to a notable broadening of the temporal correlation width of the photon pair. Consequently, the maximum attainable distance for quantum communication becomes limited, ultimately constraining the achievable key rate [17]. These correlated photon pairs birthed through SPDC, underpin various quantum techniques, including entanglement-based quantum key distribution protocols and entanglement-based clock synchronization [18–20]. Their robust temporal correlation renders them indispensable components within the realm of quantum technologies [21–24].
In practical applications of SPDC, the management of spectral dispersion is facilitated by the inherent broadband nature of the phenomenon. Fiber-based quantum technologies that rely on SPDC routinely employ spectral filtering techniques like the Bragg grating structure and spectral dispersion compensation to effectively address this issue [17,24]. Gratings, as optical devices, exhibit diverse functionalities depending on the regime in which they operate. These functionalities are contingent on several key factors, including the grating period, the amplitude of refractive index modulation, and the wavelength of incident light. Here's an overview of the various operating regimes of gratings and their associated applications [25–29]. These are diffraction, reflection, and subwavelength regimes. In the diffraction regime applications such as beam splitting, light filtering, and spectrometry are considered. In the reflection regime applications such as feedback lasers, optical filters, and sensors were studied [28–31]. Finally, in the subwavelength regime applications such as waveguides and resonators were considered. The utility of gratings is highly dependent on the specific regime in which they operate, and their versatility makes them invaluable components in a wide range of optical systems and applications [32].
In the realm of quantum structures employing entangled photons, it becomes paramount to investigate the preservation of their quantum characteristics within intricate optical circuits. The entanglement exhibited by two photons confers distinct advantages that significantly enhance the value of quantum computing over classical computing. Consequently, it is imperative to conduct thorough examinations of correlation levels in structures like Bragg gratings, which find a multitude of applications in optical integrated circuits. Such investigations should encompass a comprehensive analysis of the influence of physical parameters on the correlation function. Superposition in Bragg gratings is a technique that leverages the interference of ultraviolet beams to induce a periodic variation in the refractive index of the fiber core [30,33,34]. This variation effectively functions as a frequency-selective mirror, selectively reflecting certain wavelengths of light while permitting the transmission of others. On a related note, optical metamaterials represent a class of artificially engineered structures endowed with unique physical properties not naturally occurring in nature. These structures harness resonant elements to provoke substantial alterations in the behavior of light over distances akin to the wavelength of light in free space. This remarkable capability enables the creation of exceedingly thin metal surface components [35–40]. Furthermore, at the sub-wavelength scale, non-resonant dielectric structures serve as homogeneous media characterized by an anisotropic refractive index tensor [41]. These structures play a pivotal role in manipulating light and paving the way for a host of innovative applications and breakthroughs in optics.
In this study, our primary objective is to delve into the quantum functionality exhibited by a superimposed Bragg grating and, in doing so, discern the crucial factors that may potentially influence the correlation function of two entangled photons as they traverse this medium, both individually and simultaneously. Our overarching goal in this endeavor is to preserve the quantum nature, an indispensable aspect of quantum information processing. Entanglement, as we conceive it, empowers the transmission of information from one point to another, especially in circumstances characterized by uncertainty. This phenomenon bestows a substantial acceleration to computational speeds in quantum devices. Quantum computing protocols necessitate adept optical management to mitigate photon losses. Consequently, the preservation of quantum properties, including correlation functions and controls, stands as a pivotal consideration in the realm of optical circuits. The structure of this article is structured into three distinct sections. In the following segment, we will explore the mathematical relationship underpinning two entangled photons. Subsequently, we will embark on an in-depth discussion elucidating how a prominent medium such as the superposition Bragg grating has the potential to expand the correlation function, thereby posing a threat to the entanglement of these two photons. Lastly, we will employ numerical analysis and simulation techniques to construct correlation and scattering graphs, enabling us to draw a well-founded conclusion and compare our findings with existing knowledge.
2. Mathematical formalism
2.1. Dispersion management of superimposed Bragg grating
Bragg gratings possess a unique capability wherein their reflectivity and group delay can be intentionally influenced by manipulating their dispersion characteristics. This property finds valuable applications in countering chromatic dispersion during optical fiber communication or enhancing the precision and efficiency of optical spectroscopy. To thoroughly analyze the dispersion properties inherent in Bragg gratings, it is imperative to scrutinize both their reflection spectrum and group delay response. These parameters provide critical insights into how light is reflected and delayed within the grating structure, which is essential for tailoring their performance to specific applications [34]. The creation of a superimposed Bragg grating structure involves the amalgamation of multiple simple grating structures, each with its periodicity [31]. Figure 1 visually illustrates the process by which a superimposed Bragg grating structure is crafted. The numerical values employed in this article have been sourced from Ref. [42], ensuring accuracy and consistency in our analysis and simulations.
Fig. 1. Superposition Bragg grating structure comprises several simple Bragg gratings, (a) each with a different refractive index. (b)This work uses two periods: one with a length of 1$\mu m$, and one with a length of 3$\mu m$.
When an electromagnetic wave with a propagation vector (β) is coupled to this structure, it experiences a fluctuating refractive index within each period, leading to the reflection of certain waves. Consequently, the effective refractive index within this structure plays a pivotal role in guiding and manipulating the propagation of waves through it. This phenomenon is vividly illustrated in the figure. To determine the refractive index of a superimposed Bragg grating structure, one can utilize the Fourier series representation of the constituent structures (Eq. (1)). This mathematical approach allows for the accurate calculation of the refractive index profile within the superimposed Bragg grating, facilitating a comprehensive understanding of its optical behavior and enabling precise control over its performance characteristics.
The forward propagation constants: ${\beta _a}$
The backward propagation constants: ${\beta _b}$
The amplitude of the electric field of traveling waves: ${A_n}$
The amplitude of the electric field of the returning waves: ${B_n}$
Coupled propagation mode in the structure: ${\xi _n}$
In the context of a waveguide with single-mode property, the mode profile that propagates can be expressed as:
To obtain the dispersion relation, we need to solve Maxwell's equations for a superimposed Bragg grating structure. After solving this, we will obtain the following relations.
In this section, we have analyzed two structures characterized by two and three periodic periods, respectively. Figure 2 visually presents the graph that illustrates the mathematical equation representing the diversity of these structures. This equation was derived as a solution through numerical calculations facilitated by the utilization of MATLAB software.
To verify the accuracy of the calculations, we created a dispersion curve plot based on Ref. [41] for a periodic structure (Fig. 3). The results were consistent.
Fig. 2. Dispersion plot for (a) two-period (${g_1} = 1\mu m,\; {g_2} = 3\mu m$) and (b) three-period SM structures (${g_1} = 1\mu m,\; {g_2} = 3\mu m,{g_3} = 5\mu m$). (Matched by phase of $2\pi $, $({{\beta_a} + {\beta_b} = 2\pi } )$)
Fig. 3. A dispersion plot is used to confirm the relationship between the Bragg grating structure with the Superposition Bragg grating based on reference data from [41]
2.2. Correlation function theory
The Schrödinger equation is used to determine the initial quantum states of down-converted photons and can be expressed as follows [46]
The interaction Hamiltonian, represented by Hint, is proportional to the crystal's volume, the electric field of the pumped waves, and the induced polarization. (${H_{int}}(t )= \smallint {P_i}{E_{pump}}dV$). According to the creation and annihilation operators, the quantum state is obtained from the classical field [47,48].Let us assign ${h_{th}}({x,y} )\; $ and $S({x,y} )$ as the mode shape and cross-sectional area of the crystal, respectively. To make the process of obtaining these relationships easier, we will imagine certain conditions for the crystal where its length in the direction of photon propagation is limited and equal to Lz.
To calculate the second-order correlation function, we need to consider two positive frequency parts of the electric field operator (Es+ and Ei+). These can be derived by using Eq. (15)'s state vector and an approximated wave vector dispersion relation. Dispersion coefficients BJ0 through BJn represent orders 0 through n. ${\Delta _s} = {\nu ^2}({\beta_i^2 + \beta_s^2} )$ represents degenerate type-I SPDC, and ${\Delta _s} = \nu ({\beta_i^1 - \beta_s^1} )$ represents nondegenerate type-I SPDC [49].
As per the information provided in Ref. [46], the following relationship has been established.
The full width at half maximum is measured about [45,46]
When the first photon's positive dispersion is counteracted by the second photon's negative dispersion, a non-local dispersion cancellation effect occurs. This relationship only occurs when the two photons are entangled.
3. Result and discussion
Preserving the quantum nature within photonic substrates constitutes a critical endeavor for the advancement of quantum computing and optical quantum information processing. Optical integrated circuits stand as a promising avenue for achieving these goals, offering stability in integration, temperature operation, and compact dimensions. Among the key components within optical integrated circuits, grating structures hold significant value due to their unique characteristics. In light of this, our study is designed to elucidate the impact of the superimposed Bragg grating structure on the degree of correlation exhibited by two entangled photons. We aim to comprehensively investigate and report on the factors influencing the correlation function of these entangled photons as they traverse various environments. To achieve this, we have considered three distinct scenarios. The first one is a nondispersive medium such as air. In this scenario, both photons travel through a non-dispersive medium, which, in this case, is air. The second one is a dispersive medium. In the second scenario, both photons pass through a dispersive medium known as Bragg Grating, which introduces dispersion effects. Finally, the third one is a superimposed Bragg grating dispersive medium. Lastly, both photons traverse a superimposed Bragg dispersive medium, as illustrated in Fig. 4–7. This scenario combines the features of superposition Bragg gratings with dispersion characteristics. The structures considered in our simulations encompass various features that contribute to their unique behavior and impact on the correlation function of entangled photons. These features and their interplay within the different scenarios will be thoroughly explored and analyzed in our study.
Fig. 4. Comparison of three different modes in the correlation function. When both entangled photons pass through air, the Bragg grating structure or Superposition Bragg grating structure. $({L = 100\mu m,\; \Lambda = 10\mu m\; ,\; {\beta_a} + {\beta_b} = 2\pi } )$.
Fig. 5. To compare the correlation of biphoton in various dispersion environments. $({L = 100\mu m,\Lambda = 10\mu m,\; {\beta_a} + {\beta_b} = 2\pi } )$
Fig. 6. Compensation of dispersion effect through the difference of effective wavelengths of two environments $({L = 100\mu m,\; {\beta_a} + {\beta_b} = 2\pi } )$
Fig. 7. Investigating how the refractive index of superposition Bragg gratings affects correlation function.$({L = 100\mu m,\Lambda = 10\mu m,\; {\beta_a} + {\beta_b} = 2\pi } )$
Bragg grating structure: ${n_f} = 3.6,\; {n_c} = 3.4,\; \Lambda = 10\mu m$
Superimposed Bragg grating structure: ${n_f} = 3.6,\; {n_{c1}} = 3.4,\; {g_1} = 1\mu m,\; \; {n_{c2}} = 3.5,\; {g_2} = 3\mu m$
The prediction that the correlation between two photons dissipates more rapidly due to the dispersion properties of the superimposed Bragg grating structure aligns with our expectations. This phenomenon clarifies why the graph depicting the correlation function in this particular case exhibits a broader profile when compared to the other two scenarios. The diagram below offers a comprehensive examination of the correlation function when both photons are subjected to different scenarios. In the first scenario, both photons are propagated through the dispersive medium (Bragg grating). In the second case, both photons are propagated through the superimposed Bragg grating dispersive medium. Finally, one of the photons is propagated through normal Bragg grating dispersive media but the second photon propagates through the superimposed Bragg grating dispersive medium. This diagram provides a comprehensive view of how the correlation function of entangled photons behaves under these different conditions, offering insights into the impact of dispersion and structural variations on their quantum properties.
The diagram indeed underscores the significance of guiding one of the entangled photons, typically referred to as the signal photon, through a superposition Bragg grating structure. This process results in a gradual loss of its quantum nature, aided by the other photon, known as the idler. Such a transformation is instrumental in optical quantum calculations within integrated circuits. In the proposed scenario, where both entangled photons (the biphoton) pass through the superimposed Bragg grating structure under the worst-case conditions, it is possible to control and even halt the spread of the correlation function by introducing a difference in their effective wavelengths (${\lambda _{eff}} = \lambda \Lambda $). As depicted in the graph, the correlation function's width can be manipulated by increasing the difference in effective wavelengths between the two environments. Essentially, the greater the difference in effective wavelengths, the narrower the correlation function becomes. This control over the correlation function width has significant implications for tailoring and optimizing quantum behavior in integrated circuits and quantum information processing applications.
To investigate how the difference in the refractive index of the gratings influences the correlation function, we subjected an idler photon to two environments characterized by distinct refractive indices. These environments exhibit the following traits.
Case 1: Superimposed Bragg grating, refractive index: ${n_f} = 3.6,\; {n_{c1}} = 3.4,\; {g_1} = 1\mu m,\; \; {n_{c2}} = 3.5,\; {g_2} = 3\mu m$
Case 2: Superimposed Bragg grating, refractive index: ${n_f} = 3.8,\; {n_{c1}} = 1.46,\; {g_1} = 1\mu m,\; \; {n_{c2}} = 1.56,\; {g_2} = 3\mu m$
By systematically varying the refractive indices (cases 1 and 2) and examining how these differences impact the correlation function between the signal and idler photons, we can gain valuable insights into the role of refractive index variations in modulating quantum behavior and entanglement in optical systems.
As the diagram indicates, the biphoton's passage through two superimposed Bragg grating structures with differing refractive indices leads to a broadened correlation function. To optimize results and maintain a narrower correlation function, the photons should traverse structures with identical refractive indices. To create a superimposed Bragg grating structure with variations in refractive index, we employ three periods, as illustrated in Fig. 8. The previous section presented the dispersion diagram for this specific structure, shedding light on its dispersion characteristics and behavior concerning varying refractive indices.
The correlation function is expected to be wider in the superimposed Bragg grating structure with three periods due to increased scattering. This observation is illustrated in Fig. 9.
Fig. 9. Comparison of the correlation function in biphoton motion inside Bragg grating with 2- and 3-periods.$({L = 100\mu m,\Lambda = 10\mu m,\; {\beta_a} + {\beta_b} = 2\pi } )$
Throughout this article, we have delved into a comprehensive exploration of the influence of various components of the superimposed Bragg grating structure on the correlation function of entangled photons. This investigation carries significant implications for the field of optical quantum information processing, where entangled photons play a pivotal role in the remote control of quantum states. These advancements hold the potential to drive breakthroughs in quantum computing and photonics. The article underscores the paramount importance of preserving the quantum properties of the system and effectively compensating for the dispersion effects that can impact quantum behavior. A thorough and insightful analysis of the superposition Bragg grating structure has been meticulously presented, shedding light on its intricate dispersion characteristics and their implications for the entanglement and control of photons.
4. Conclusion
In the realm of quantum computing, the preservation of the quantum nature of optical signals traversing integrated circuits stands as a critical endeavor. However, the dispersion properties of the media within these circuits can introduce challenges, potentially leading to the degradation of quantum properties over time. In response to this issue, our research has focused on a specific category of grating structures known as superimposed Bragg gratings. These structures are commonly employed for filtering and coupling applications within integrated circuits. Through a meticulous exploration of the factors contributing to the loss of quantum properties in these structures, we aim to enhance our ability to manage and maintain their functionality. This research leverages the behavior of two entangled photons that traverse distinct environments. The degree of correlation observed between these photons serves as a key indicator of the structure's ability to uphold its quantum properties. By manipulating one of the entangled photons, we can effectively counteract the impact of factors like dispersion on the correlation function, thereby preventing it from spreading out. The study of the superimposed Bragg grating structure encompasses various scenarios, including cases where one photon is in the air while the other resides within the grating. Additionally, it investigates the influence of multiple gratings, discrepancies in refractive indices, and variations in effective wavelengths. To ensure the preservation of the quantum nature of optical substrates for information processing, our research proposes essential adjustments aimed at mitigating the dispersion effect on the correlation function. The significance of quantum information processing lies in its capability to accelerate complex simulations that would be infeasible to execute using classical computing methods. This research endeavors to contribute to the advancement of quantum computing and its potential for groundbreaking computational capabilities.
Disclosures
The authors have no conflicts of interest to declare relevant to this article’s content.
Author Contributions: S. A. simulated the work, and wrote the original draft. A.R. conceived the basic idea, supervised the project, and wrote and revised the manuscript.
Data availability
Not applicable
References
1. A. Sit, Frédéric Bouchard, Robert Fickler, et al., “High-dimensional intracity quantum cryptography with structured photons,” Optica 4(9), 1006–1010 (2017). [CrossRef]
2. L. Shi, E.J. Galvez, and R.R. Alfano, “Photon entanglement through brain tissue,” Sci. Rep. 6(1), 37714 (2016). [CrossRef]
3. P. Shadbolt, Jonathan C. F. Mathews, Anthony Laing, et al., “Testing foundations of quantum mechanics with photons,” Nat. Phys. 10(4), 278–286 (2014). [CrossRef]
4. A. Büse, Mathieu L. Juan, Nora Tischler, et al., “Symmetry protection of photonic entanglement in the interaction with a single nanoaperture,” Phys. Rev. Lett. 121(17), 173901 (2018). [CrossRef]
5. J. Lasa-Alonso, Martín Molezuelas-Ferreras, J J Miguel Varga, et al., “Symmetry-protection of multiphoton states of light,” New J. Phys. 22(12), 123010 (2020). [CrossRef]
6. J.M. Varga, Jon Lasa-Alonso, Martin Molezuelas-Ferreras, et al., “Bandwidth control of the biphoton wavefunction exploiting spatio-temporal correlations,” Opt. Commun. 504, 127461 (2022). [CrossRef]
7. A. Joobeur, B.E. Saleh, and M.C. Teich, “Spatiotemporal coherence properties of entangled light beams generated by parametric down-conversion,” Phys. Rev. A 50(4), 3349–3361 (1994). [CrossRef]
8. R. de J León-Montiel, J Svozilík, L J Salazar-Serrano, et al., “Role of the spectral shape of quantum correlations in two-photon virtual-state spectroscopy,” New J. Phys. 15(5), 053023 (2013). [CrossRef]
9. J.M. Lukens, Amir Dezfooliyan, Carsten Langrock, et al., “Orthogonal spectral coding of entangled photons,” Phys. Rev. Lett. 112(13), 133602 (2014). [CrossRef]
10. A. Sergienko, M.C. Booth, A.V. Sergienko, et al., “Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion,” 2005.
11. A. Valencia, Alessandro Ceré, Xiaojuan Shi, et al., “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99(24), 243601 (2007). [CrossRef]
12. D. Kielpinski, J.F. Corney, and H.M. Wiseman, “Quantum optical waveform conversion,” Phys. Rev. Lett. 106(13), 130501 (2011). [CrossRef]
13. A. Dosseva, Ł. Cincio, and A.M. Brańczyk, “Shaping the joint spectrum of down-converted photons through optimized custom poling,” Phys. Rev. A 93(1), 013801 (2016). [CrossRef]
14. A. Chiuri, I. Gianani, V. Cimini, et al., “Ghost imaging as loss estimation: Quantum versus classical schemes,” Phys. Rev. A 105(1), 013506 (2022). [CrossRef]
15. S.P. Neumann, Thomas Scheidl, Mirela Selimovic, et al., “Model for optimizing quantum key distribution with continuous-wave pumped entangled-photon sources,” Phys. Rev. A 104(2), 022406 (2021). [CrossRef]
16. H. Hong, Runai Quan, Xiao Xiang, et al., “Demonstration of 50 km Fiber-optic two-way quantum time transfer at femtosecond-scale precision,” arXiv, arXiv:2111.00380 (2021). [CrossRef]
17. S. Wengerowsky, Siddarth Koduru Joshi, Fabian Steinlechner, et al., “Entanglement distribution over a 96-km-long submarine optical fiber,” Proc. Natl. Acad. Sci. 116(14), 6684–6688 (2019). [CrossRef]
18. J. Nunn, L. J. Wright, C. Söller, et al., “Large-alphabet time-frequency entangled quantum key distribution using time-to-frequency conversion,” Opt. Express 21(13), 15959–15973 (2013). [CrossRef]
19. F. Hou, Runai Quan, Ruifang Dong, et al., “Fiber-optic two-way quantum time transfer with frequency-entangled pulses,” Phys. Rev. A 100(2), 023849 (2019). [CrossRef]
20. J.-N. Zhang, Yang Ran, Li Xinhui, et al., “Realization of a source-device-independent quantum random number generator secured by nonlocal dispersion cancellation,” Adv. Photonics 5(03), 036003 (2023). [CrossRef]
21. Y. Shi, Soe Moe Thar, Hou Shun Poh, et al., “Stable polarization entanglement based quantum key distribution over a deployed metropolitan fiber,” Appl. Phys. Lett. 117(12), 1 (2020). [CrossRef]
22. J. Lee, Lijiong Shen, Alessandro Cerè, et al., “Symmetrical clock synchronization with time-correlated photon pairs,” Appl. Phys. Lett. 114(10), 1 (2019). [CrossRef]
23. C. Lee, Darius Bunandar, Zheshen Zhang, et al., “High-rate large-alphabet quantum key distribution over deployed telecom fiber,” in CLEO: QELS_Fundamental Science. 2016. Optica Publishing Group.
24. R. Chua, J.A. Grieve, and A. Ling, “Fine-grained all-fiber nonlocal dispersion compensation in the telecommunications O-band,” Opt. Express 30(9), 15607–15615 (2022). [CrossRef]
25. H. Heidarzadeh, Ali Rostami, Mahboubeh Matloub, et al., “Analysis of the light trapping effect on the performance of silicon-based solar cells: absorption enhancement,” Appl. Opt. 54(12), 3591–3601 (2015). [CrossRef]
26. A. Rostami, S. Zirak Gharamaleki, H. Alipour Banaei, et al., “Multi-channel narrowband DWDM optical filters based on generalized aperiodic Thue-Morse structures,” in 2009 International Symposium on Optomechatronic sTechnologies. 2009. IEEE.
27. S. Golmohammadi, Y. Rouhani, Karim Abbasian, et al., “Photonic bandgaps in quasiperiodic multilayer structures using Fourier transform of the refractive index profile,” Prog. Electromagn. Res. B 18, 311–325 (2009). [CrossRef]
28. N. Asadian, S. Matloub, and A. Rostami, “Two-color terahertz Radiation Emission in Quasi-Periodic Smith–Purcell Structures,” IEEE Trans. Plasma Sci. 47(7), 3344–3351 (2019). [CrossRef]
29. A Rostami, “Optical filters using homogeneous layered media with a solvable index of refractions,” Laser Phys. 12(11), 1358–1362 (2002).
30. S. Golmohammadi and A. Rostami, “Optical filters using optical multi-layer structures for optical communication systems,” Fiber Integr. Opt. 29(3), 209–224 (2010). [CrossRef]
31. H. Ghafoori-Fard, M. Moghimi, and A. Rostami, “Linear and nonlinear superimposed Bragg grating: A novel proposal for all-optical multi-wavelength filtering and switching,” Prog. Electromagn. Res.77, 243–266 (2007). [CrossRef]
32. B.K. Park, Yong-Su Kim, Osung Kwon, et al., “High-performance reconfigurable coincidence counting unit based on a field programmable gate array,” Appl. Opt. 54(15), 4727–4731 (2015). [CrossRef]
33. A. Yazdanpanah-Goharrizi and A. Rostami, “Reconstruction of non-uniform fiber Bragg grating parameters using a combination of RBF and MLP neural networks,” in Photonics North 2006. 2006. SPIE.
34. A. Rostami, H. Rasooli, and H. Baghban, Terahertz technology: fundamentals and applications. Vol. 77. 2010: Springer Science & Business Media.
35. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014). [CrossRef]
36. J.M. Luque-González, Alejandro Sánchez-Postigo, Abdelfettah Hadij-ElHouati, et al., “A review of silicon subwavelength gratings: building break-through devices with anisotropic metamaterials,” Nanophotonics 10(11), 2765–2797 (2021). [CrossRef]
37. Q. Wang, Edward T. F. Rogers, Behrad Gholipour, et al., “Optically reconfigurable metasurfaces and photonic devices based on phase change materials,” Nat. Photonics 10(1), 60–65 (2016). [CrossRef]
38. S. Kruk and Y. Kivshar, “Functional meta-optics and nanophotonics governed by Mie resonances,” ACS Photonics 4(11), 2638–2649 (2017). [CrossRef]
39. P. Lalanne and P. Chavel, “Metalenses at visible wavelengths: past, present, perspectives,” Laser Photonics Rev. 11(3), 1600295 (2017). [CrossRef]
40. S.M. Kamali, Ehsan Arbabi, Amir Arbabi, et al., “A review of dielectric optical metasurfaces for wavefront control,” Nanophotonics 7(6), 1041–1068 (2018). [CrossRef]
41. S Rytov, “Electromagnetic properties of a finely stratified medium,” Soviet Physics JEPT 2, 466–475 (1956).
42. A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13(4), 233–253 (1977). [CrossRef]
43. M. Moghimi, H. Ghafoori-Fard, and A. Rostami, “Analysis and design of all-optical switching in apodized and chirped Bragg gratings,” Prog. Electromagn. Res. B 8, 87–102 (2008). [CrossRef]
44. A Rostami, “Generalized Fibonacci quasi photonic crystals and generation of superimposed Bragg Gratings for optical communication,” Microelectron. J. 37(9), 897–903 (2006). [CrossRef]
45. P. Ribeiro and M. Raposo, Photoptics 2015. Vol. 177. 2016: Springer.
46. A. Christ, “Probing multimode squeezing with correlation functions,” New J. Phys. 13(3), 033027 (2011). [CrossRef]
47. S.-Y. Baek, Y.-W. Cho, and Y.-H. Kim, “Nonlocal dispersion cancellation using entangled photons,” Opt. Express 17(21), 19241–19252 (2009). [CrossRef]
48. X. Xiang, Ruifang Dong, Baihong Li, et al., “Quantification of nonlocal dispersion cancellation for finite frequency entanglement,” Opt. Express 28(12), 17697–17707 (2020). [CrossRef]
49. D. Llewellyn, Yunhong Ding, Imad I. Faruque, et al., “Chip-to-chip quantum teleportation and multi-photon entanglement in silicon,” Nat. Phys. 16(2), 148–153 (2020). [CrossRef]
