Wavelength-scale optical parametric oscillator

Abstract

An OPO includes a resonator comprising a material with nonlinear susceptibility that generates an output electromagnetic field in response to a pump electromagnetic field input to the material, the output electromagnetic field having one or more output wavelengths that are longer than one or more pump wavelengths of the pump electromagnetic field, and the resonator has dimensions in free space that are less than or on the order of the one or more output wavelengths.

Description

Cross - Reference to Related Applications

[0001] This application claims the benefit of U.S. Provisional Application Serial No. 62 / 932,647, filed on November 8, 2019, by Saman Jahani and Alireza Marandi, titled "WAVELENGTH SCALE OPTICAL PARAMETRIC OSCILLATORS", and naming the same inventors, and client reference CIT - 8388 - P, which is incorporated herein by reference. Statement Regarding Federally Sponsored Research and Development

[0002] The present invention was made with government support under award number W911NF - 18 - 1 - 0285, issued by the Army. The government has certain rights in this invention. Background of the Invention

[0003] 1. Technical Field The present invention relates to an Optical Parametric Oscillator (OPO) and a method of making the same.

[0004] 2. Description of Related Art (Note: Throughout this application, a number of different publications are referenced, as indicated by one or more reference numbers within brackets, e.g., [x]. A list of these different publications, arranged according to the reference numbers, is in the section titled "References" below. Each of these publications is incorporated herein by reference.)

[0005] Optical parametric oscillators (OPOs) are widely used in applications ranging from metrology and spectroscopy to quantum information science [12, 56, 4, 32, 24, 46, 20, 43, 52]. An OPO consists of a medium with a second- or Kerr-nonlinearity inside a resonator, typically much larger than the operating wavelength, which converts pump photons into signal and idler photons [56, 32, 24, 46, 20, 4]. In degeneracy, the indistinguishable signal and idler of the OPO can form a squeezed vacuum state below the oscillation threshold [36, 61] that is used in several applications in quantum information processing [52, 8, 41, 44]. Above threshold, the conversion efficiency rises rapidly and the output signal exhibits a two-level phase state that can be utilized as spins in artificial Ising networks [33, 35]. Degenerate OPOs above threshold are also effectively used for the generation of optical frequency combs in the mid-infrared regime [32, 43].

[0006] Recent progress in nanoscale optical confinement and precise nanofabrication of challenging nonlinear materials [[59, 31]] have inspired the idea of rethinking the potential for miniaturization up to the limits of nonlinear systems. Miniature OPOs have recently been demonstrated based on Kerr nonlinearities [[46, 24, 10]] and second-order nonlinearities [[5]], and on-chip OPOs based on whispering gallery resonators [

[60] ]. The size of these resonators is still several orders of magnitude larger than their operating wavelengths. Confinement of strong fields inside nanostructures has revealed the potential of nonlinear optics at the nanoscale [[57, 21, 40, 45, 50, 63]]. However, the main focus so far has been on upconversion in nanostructures, and optical parametric oscillation in wavelength-scale structures has remained unexplored. Conventional theories that have mainly evolved for traveling-wave nonlinear optical systems [

[17] ] or high-Q resonators [[19, 11]] cannot accurately model OPOs in nanostructures even when applied as is. The reason is that spatial variations of the field occur in the subwavelength regime where the slowly varying envelope approximation (SVEA) is no longer valid [

[17] ]. Furthermore, unlike conventional large OPOs, in nanostructure resonators, the input pump can excite several modes of the cavity at the pump wavelength, and due to the low Q of the modes, the pump can also directly interact with several modes at the signal wavelength. In recent years, a few theoretical models have been proposed to explain spontaneous downconversion in Mie resonators [

[48] ] and thresholds in 2D material-based OPOs [[9]]. However, these theories are either limited to specific structures or cannot explain the behavior of systems beyond the threshold. Furthermore, conventional designs and theories do not explain how to achieve optical parametric oscillation in wavelength-scale resonators. This disclosure meets this need. SUMMARY OF THE INVENTION

[0007] The present disclosure discloses a sub-wavelength and wavelength-scale optical parametric oscillator (OPO), predicts the behavior of the sub-wavelength and wavelength-scale optical parametric oscillator (OPO), and estimates the oscillation threshold of the sub-wavelength and wavelength-scale optical parametric oscillator (OPO). We also demonstrate a clear correlation between the second harmonic generation efficiency and the OPO threshold. This enables the estimation of the OPO threshold based on the second harmonic generation measured or simulated in different classes of resonators, such as bound states in the continuum and inverse design resonators. Our approach for analyzing and modeling small OPOs provides unprecedented opportunities for classical and quantum nonlinear photonics.

[0008] The devices disclosed herein can be embodied in many ways including, but not limited to, the following. 1. One or more optical parametric oscillators (OPOs), each comprising a resonator that includes a material having a nonlinear susceptibility to generate an output electromagnetic field in response to a pump electromagnetic field input to the material, the output electromagnetic field having one or more output wavelengths that are longer than one or more pump wavelengths of the pump electromagnetic field, the resonator being a device having dimensions less than one or more output wavelengths or on the order of one or more output wavelengths in free space. 2. The device of Example 1, wherein the resonator comprises particles having dimensions. 3. The device of Example 1 or 2, wherein the resonator supports one or more plasmon modes of at least one of the pump electromagnetic field or the output electromagnetic field. In one or more embodiments, the resonator supports a plasmon that confines the pump and / or output in the resonator. 4. The device of any of Example 1 or 3, wherein the resonator comprises a structure having a gap that supports a plasmon mode that overlaps with the material. 5. The device of Example 4, further comprising an additional material having a second-order nonlinear susceptibility over the gap. 6. The resonator comprises an additional substance having optical properties (e.g., relative permittivity) different from those of the substance, the resonator having the additional substance and the substance, adapted to the first interaction of the pump electromagnetic field and the second interaction of the output electromagnetic field, and having a structure consisting of the first interaction of the pump electromagnetic field and the second interaction of the output electromagnetic field, such that the efficiency of the parametric interaction between the pump electromagnetic field and the output electromagnetic field is increased as compared to the case where there is no additional substance, the device of any one of Examples 1 to 5. 7. The resonator comprises at least one of a polymer, glass, a linear substance, or a refractive index of less than 2, or comprises an additional substance consisting essentially of at least one of a polymer, glass, a linear substance, or a refractive index of less than 2, the device of any one of Examples 1 to 6. In an example, the linear substance is defined as having no second-order susceptibility. In another example, the linear substance is defined as a substance that is "non - linear". 8. The additional substance comprises a polymer, the device of any one of Examples 5 to 7. 9. The resonator is arranged to optimize the overlap between the pump electromagnetic field and the output electromagnetic field, or to match the overlap between the pump electromagnetic field and the output electromagnetic field, and / or to reduce the oscillation threshold of the OPO, and comprises a plurality of regions or pixels having different relative permittivities and thicknesses, the device of any one of Examples 1 to 8. 10. The largest of the dimensions is less than 10 microns, or the resonator fits within a sphere having a radius of 5 microns, the device of any one of Examples 1 to 9. 11. The resonator supports one or more quasi - normal electromagnetic modes of the pump electromagnetic field and / or quasi - normal electromagnetic modes of the output electromagnetic field, the device of any one of Examples 1 to 10. 12. The quasi - normal electromagnetic mode comprises one or more multipole Mie resonances comprising the output electromagnetic field, the device of Example 11. 13. Due to the low Q - factor of the multipole modes in a wavelength - scale resonator, a non - linear interaction occurs between these modes, the device of Example 12 based on Mie - type multipole resonance. As a result, the OPO threshold can be reduced by a factor significantly larger than the number of interacting modes as compared to the single - mode case. 14. Any device of an embodiment having multimode interactions leading to a phase transition manifested through parametric gain and abrupt changes in the oscillation threshold. 15. Any device of embodiments 1 to 14, further comprising a disk, cylinder, or sphere including a resonator. 16. Any device of embodiments 1 to 15, wherein the resonator has a polygonal cross-section or any cross-section (e.g., circular or non-uniform cross-section). 17. Any device of embodiments 1 to 16, wherein the resonator is patterned by lithography. 16. Any device of embodiments 1 to 17, wherein the material comprises at least one of a metal, dielectric, semiconductor, or polymer. 19. Any device of embodiments 1 to 18, wherein the material has at least one of a second-order susceptibility or a third-order susceptibility. 20. Any device of embodiments 1 to 19, comprising a plurality of OPOs, wherein the resonators are coupled evanescently or through a waveguide or an auxiliary cavity. 21. Any device of embodiments 1 to 20, comprising a plurality of OPOs that output a plurality of output electromagnetic fields in response to a plurality of pump electromagnetic fields, each of the pump electromagnetic fields having at least one of a phase or amplitude different from the phase or amplitude of another pump electromagnetic field among the pump electromagnetic fields. 23. A sensor comprising a network comprising a plurality of OPOs of any of embodiments 1 to 19 and one or more detectors coupled to detect an output electromagnetic field, and sensing a pump electromagnetic field or an environment near the network through detection of the output electromagnetic field by the detector. 23. A network comprising a plurality of OPOs of claim 1 and a coupling between the OPOs, the coupling being adjusted to model an array of coupled spins such that the minimum threshold of each of the OPOs corresponds to one minimum energy configuration of the coupled spins in the array and comprising an optical computer. 24. A device according to any of Examples 1 to 23 having a multimode interaction leading to a phase transition manifested through parametric gain and abrupt changes in the oscillation threshold that can be utilized for enhancing perception.

[0009] The present disclosure further discloses a method of operating an OPO according to one or more of Examples 1 to 21, including the steps of inputting a pump electromagnetic field into a resonator and configuring the OPO for at least one of the following: An OPO operating in degeneracy, wherein at least one of the output wavelengths is twice at least one of the pump wavelengths, An output electromagnetic field comprising an optical frequency comb including a series of equally spaced frequency peaks, An output electromagnetic field having an output spectrum wider than the input spectrum of the pump electromagnetic field at a frequency unit measured at a level of 30 dB below the peak, or The pump electromagnetic field comprises a continuous wave, a time-varying, or a pulsed electromagnetic field. Brief Description of the Drawings

[0010] JPEG0007702146000001.jpg233166JPEG0007702146000002.jpg239166JPEG0007702146000003.jpg238166 Modes for Carrying Out the Invention

[0011] In the following description of the preferred embodiments, reference is made to the accompanying drawings, which form a part hereof, and in which are shown by way of illustration specific embodiments in which the invention may be practiced. It is to be understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the present invention.

[0012] Technical Description This disclosure describes general conditions for parametric oscillation in sub - wavelength and wavelength - scale resonators. In the low - Q regime of these resonators, multiple modes near the signal wavelength can spectrally and spatially overlap (Figure 1). This enables them to interact non - linearly with each other through the pump.

[0013] In a first example, the OPO threshold in a nanostructure (AlGaAs nanoparticles) that supports Mie - type multipole resonances is estimated. It is shown that multimode interactions at the signal wavelength can lead to a significant reduction in the threshold by a factor that is significantly higher than the number of modes. Multimode interactions, as a result, also bring about a phase transition from degeneracy to non - degeneracy in these resonators, along with parametric gain and / or a sharp change in the oscillation threshold that can be utilized for ultra - sensitive measurements. Furthermore, the correlation between the up - conversion process in the nanostructure and parametric down - conversion is demonstrated. This enables the definition of the parameter space of an OPO that manipulates sum - frequency / second - harmonic generation. However, the approach described herein is general and can predict optical parametric oscillation in a wide range of resonators such as bound states in the continuum, photonic crystals, inverse - designed cavities, plasmonic resonators, and various other nanostructures and micro - structure resonators.

[0014] JPEG0007702146000004.jpg95166

[0015] JPEG0007702146000005.jpg64166

[0016] JPEG0007702146000006.jpg119166

[0017] 2. Example Results Our model is general and applicable to a wide range of resonators. First, we apply our model to estimate the threshold in an AlGaAs sphere (Fig. 2A). The reason for choosing this simple structure is that the eigenmodes of this structure can be derived analytically and represented as multipole resonances [22, 27]. Since the modes of a wide range of nanostructures such as cylinders and cubes can also be represented as multipole resonances, our results can somewhat clarify the possibility of OPOs with similar structures more suitable for on-chip fabrication [3, 26, 47, 30, 7, 34, 58, 25, 14]. Moreover, AlGaAs has JPEG0007702146000007.jpg12127 and, at optical frequencies JPEG0007702146000008.jpg1783 is a material

[15] , and with appropriate orientation [6], strong second-harmonic generation at the nanoscale has been investigated in recent years [15, 57, 58, 34, 25]. Therefore, AlGaAs has a relatively low threshold and is an excellent candidate for demonstrating OPOs at the wavelength scale. For a general dispersive or non-spherical three-dimensional resonator (Fig. 1A), the Lorentz reciprocity theory can be used to find the quasi-normal modes of the resonator [55, 49, 28, 64]. Details are reported in Section 3.

[0018] Fig. 2B shows the normalized scattering coefficients of the first six electric and magnetic modes of a particle with a radius of 500 nm. When the particle is excited by a plane wave (or Gaussian beam), several multipole modes are excited. First, set the pump in the sub-wavelength regime (pump wavelength > 1500 nm) where the first two electric and the first two magnetic modes can be excited at the signal and idler frequencies. Subsequently, consider the OPO behavior in the wavelength scale regime (pump wavelength ≈ 1000 nm) where higher-order modes can also contribute.

[0019] When operating in the sub-wavelength regime (i.e., the pump wavelength is larger than the particle size), only the first two electric and the first two magnetic modes can oscillate in the down-conversion process. The higher-order modes have large detuning JPEG0007702146000009.jpg can be ignored. The electric field profiles of these four modes are shown in Fig. 2C. The contribution of each mode in the OPO signal / idler supermode is determined by the overlap of the field between the pump and the mode, the inter-mode nonlinear coupling as represented by Equation 2, the Q value, and the detuning from the second harmonic frequency. Figs. 3A - 3B show the oscillation threshold and the spectral separation of the signal and idler as a function of the pump wavelength. The decrease in the threshold spectrum near 1830 nm is due to the enhancement of the pump field as a result of the excitation of the three magnetic modes. Even away from the center of resonance, the input pump can excite a number of low-order modes of the resonator.

[0020] If we assume that inter-mode coupling can be ignored and only one of the eigenmodes can oscillate, the OPO threshold becomes significantly higher. For example, when the pump is at the center of the three magnetic resonance, the minimum threshold of the single-mode OPO is around 0.27 MW, which is 36 times higher than the threshold shown in Fig. 3A where multi-mode interactions are considered (see Section 3 for the thresholds and coupling coefficients of all modes). In a traveling-wave multi-mode OPO, in the best scenario, it is understood that the threshold is of the order of the single-mode threshold divided by the number of modes

[11] . The reason is that the modes in a traveling-wave resonator have the same nature. Thus, the maximum overlap is achieved if all modes have the same mode profile [11, 1]. However, in a wavelength-scale OPO, each of the multipole modes has a different spatial distribution, and their overlap through the pump field can potentially lead to JPEG0007702146000010.jpg even stronger coupling than 1464.

[0021] As shown in FIGS. 3A - 3B, when the OPO undergoes a transition from non - degenerate to degenerate oscillation, a sharp drop in the threshold occurs. This corresponds to the disorder - to - order phase transition demonstrated in traveling - wave OPOs in recent years

[54] . To understand the phase transition in the wavelength - scale OPO, it is necessary to look at the eigenvalues and eigenvectors of these resonators. For example, focus on the case of degeneracy with pump excitation at 1900 nm (FIG. 4A). FIGS. 4B and 4C show the real and imaginary parts of the eigenvalues as a function of the input power, respectively. Since the four modes are related at the signal and idler frequencies, there are eight eigenvalues and eight corresponding super - modes. The OPO threshold of each super - mode is defined when the imaginary part of the eigenvalue passes through zero (FIG. 4B).

[0022] At low input - power levels, weak coupling between the eigen - modes as seen in Equation 1 occurs. Thus, each super - mode is dominated by a single eigen - mode (see Section 3 for eigen - vectors). However, as the input power increases, the modes begin to interact due to non - linear coupling through the pump. As a result, the super - modes near and above the threshold become a superposition of all the eigen - modes. The electric - field distributions of the four oscillating super - modes at the threshold are shown in FIG. 4D.

[0023] Furthermore, due to the detuning of the resonance centers of the eigen - modes from the second - harmonic, all the signal / idler super - modes of the eigenvalues are non - degenerate at low input - power levels JPEG0007702146000011.jpg1764 (FIG. 4C). The increase in the input power strengthens the mode coupling that can change the signal and idler spectral separation. This can lead to the phase transition from non - degenerate to degenerate and vice versa. In particular, with a very strong force, the non - linear coupling dominates the detuning (Equation 1), and as a result, all the modes are synchronized at the second - harmonic frequency (FIG. 4C).

[0024] The phase transition at the maximum eigenvalue is shown in Fig. 5. This phase transition occurs simultaneously with a sharp change in the parametric gain that can be utilized for enhancing sensing and computing [54, 2, 62]. The phase transition can occur either due to the competition between the eigenvalues that achieve the highest gain or the coalescence of two eigenvalues. When the critical point is the coalescence of two eigenvalues, the eigenvectors also coalesce at the critical point (see Section 3), which is a characteristic of exceptional points in non-Hermitian systems [38, 53]. We have shown this type of first-order phase transition in a coupled OPO

[54] . However, the phase transition proposed here is observed in a single-wavelength-scale OPO due to the strong nonlinear coupling between multiple modes of the resonator.

[0025] To improve the performance of the OPO, it is desirable to further reduce the oscillation threshold. The OPO threshold is inversely proportional to the Q value of the pump mode when only one mode exists at the pump frequency (see Section 3). Therefore, as the Q value of the higher-order multipole mode becomes even higher, it is expected that the threshold will be further reduced by exciting the higher-order mode. Fig. 6A shows 10 4Shows the OPO threshold of the first oscillating supermode as a function of the pump wavelength near the 6 magnetic modes at 1110 nm with a Q value of [[ID=]], and the 5 electric modes at 1125 nm with a Q value of 2500. The separation of the signal / idler frequencies from the second harmonic is shown in Fig. 6B. For the signal and idler, all modes with resonance wavelengths longer than the pump wavelength (the first 4 electric and the first 5 magnetic modes) were considered. The electric field distribution of the pump, and the first signal / idler supermodes of magnetic and electric mode excitation are shown in Figs. 6c and 6d, respectively. The thresholds at the centers of resonance of the 6 magnetic modes and 5 electric modes can reach 2W and 460W, respectively. Due to the large signal and idler separation, the parametric gain is low. However, at an input power of 43W near the 6 magnetic modes and 1900W near the 5 electric modes, the OPO undergoes a phase transition to a degenerate regime, and the parametric gain is dramatically enhanced (see Section 3). In the case of the 5 electric modes, note that although Q is large and high-Q modes can be excited at the signal wavelength, the threshold is not very different from the sub-wavelength regime shown in Figs. 3A - 3B. This is because in the absence of phase matching in a large resonator, the field overlap between the pump and signal modes is weak. Due to the competition between different eigenvalues, a phase transition occurs in the non-degenerate regime accompanied by a sharp change in the signal / idler spectrum separation, which can lead to a discontinuous change in the derivative of the OPO threshold as shown in Fig. 6A, as seen in Fig. 6B.

[0026] The approach we used to estimate the threshold can also be applied to estimate second harmonic generation in multimode wavelength scale resonators (see Section 3 for details). Specifically, if both the pump and the signal are single-mode and the detuning from the eigenfrequency is negligible, the OPO threshold, and the second harmonic generation efficiency,

Number

[0027] There is no threshold for the SHG process, and since conventional detectors are more accurate at short wavelengths

[16] , it is usually easy to simulate or measure the SHG process. This enables the estimation of the OPO thresholds in several structures that have already been proposed for SHG. Figure 7 shows some examples and the estimated thresholds in these structures. The low threshold in the inverse design structure

[29] indicates the importance of the field overlap in achieving a strong nonlinear response. Note that the thresholds reported in Figure 7 are for the case of a continuous-wave source.

[0028] In the wavelength-scale OPO, the round-trip time is only a few femtoseconds and the Q value is relatively small compared to microresonators, so the input pump can be gradually compressed into shorter pulses. As a result, even in the sub-wavelength OPO, the average power threshold becomes several tens of milliwatts (pulse repetition rate 100 MHz), which is on the order of the threshold of a free-space pulsed OPO [32, 43]. Therefore, oscillation can occur before the material damage threshold. The field overlap can be further enhanced by using a hybrid plasmonic structure

[45] or by controlling the evanescent wave

[23] , through Mie resonance engineering and inverse design

[39] . This can potentially help to achieve sub-milliwatt oscillation thresholds in sub-wavelength and wavelength-scale resonators.

[0029] In conclusion, we have proposed a general theory for estimating the oscillation threshold and the nonlinear mixing behavior of modes above the threshold in wavelength-scale OPOs. We have shown that the nonlinear interactions in multi-mode wavelength-scale resonators can be different from their larger counterparts, and as a result of the multi-mode interactions in these resonators, the threshold can be significantly reduced. We have demonstrated a phase transition in these resonators due to the nonlinear interactions between multiple modes. We have shown that phase matching is not required in this regime, but the field overlap between modes can play an important role in reducing the threshold. Our formalism is general and can predict the behavior of OPOs above the threshold if pump light attenuation is also considered. This is JPEG0007702146000013.jpg913 can also be applied to cavities. Our approach enables the design of a new class of non-linear integrated photon systems.

[0030] 3. Equation Derivation In this section, equations for single-mode and multi-mode OPOs in both degenerate and non-degenerate cases are derived. We derive the second-harmonic generation (SHG) efficiency and demonstrate the relationship between the SHG efficiency and the threshold in degenerate OPOs for the single-mode case. We consider the role of quasi-normal modes in the case of distributed and aspherical, and the role of low-Q background modes in the performance of arbitrarily shaped OPOs. In the following sections, the parameters, eigenvalues, and eigenvectors of the presented results are communicated in more detail.

[0031] a. Wave Equation The Helmholtz wave equation with non-linear polarization can be described as follows:

Equation

Equation

[0032] First, the non-linear dynamics for single-mode OPO in degeneracy are formulated below, and then the formalism is extended to the multi-mode cavity and non-degenerate cases.

[0033] By introducing Equation 7 into Equation 6 and considering the k-th mode as the only mode at the operating frequency, we have:

Equation

[0034] JPEG0007702146000017.jpg51166

[0035] JPEG0007702146000018.jpg25166

[0036] Note that the above equation was derived assuming a weak material dispersion. In the case of a dispersive structure, a more rigorous analysis is required for the mode evolution

[64] . First, implement the nonlinear dynamics and estimate the threshold in a single-mode OPO. Subsequently, extend our model when the cavity has multiple modes at the signal wavelength. Apply our model to second-harmonic generation as well, and show that the threshold can be estimated from the SHG efficiency when the second-harmonic signal is in a single mode. This can be useful for estimating the OPO threshold of structures already proposed for SHG.

[0037] b. Second-Harmonic Generation By describing the nonlinear polarization, the nonlinear dynamics for different nonlinear processes (e.g., second-harmonic generation and second-subharmonic generation) can be found. Here, first focus on the threshold of second-subharmonic generation in a degenerate OPO. For simplicity, the modal resistive losses are ignored.

[0038] The coupled nonlinear wave equations for the signal and pump can be written as follows:

Equation

[0039] JPEG0007702146000021.jpg65166

[0040] JPEG0007702146000022.jpg75166

[0041] JPEG0007702146000023.jpg85166

[0042] JPEG0007702146000024.jpg38166

[0043] Therefore, the threshold for an input source exceeding the threshold is as follows:

Number

[0044] If there are two or more coupling channels between the input and the cavity, such as excitation from free space, Equation 19 is not accurate, and the coupling between the input power and the pump mode amplitude, JPEG0007702146000026.jpg1511 should be derived from the linear analysis of the cavity at the pump frequency.

[0045] JPEG0007702146000027.jpg40166

[0046] The steady-state response of this equation can be written in matrix form as follows:

Number

[0047] The OPO threshold is the minimum pump power at which the determinant passes through zero. Near the threshold, it is the only oscillation mode, and the eigenvector corresponding to its eigenvector represents the spatial distribution of the signal. The phase difference between each mode of the pulse and the pump is automatically set to achieve the minimum threshold. If the Q value of the mode or the center frequency of all modes is not the same, there is no closed-form solution for the eigenvalue. However, in the best scenario where all modes have similar non-linear coupling coefficients and Q values, the threshold decreases only by a factor of the number of modes.

[0048] As shown in Figures 3 and 6 in Sections 1 and 2, the threshold of the degenerate OPO is not necessarily smaller than that of the non-degenerate case. Therefore, it is extremely important to consider the non-degenerate case as well.

[0049] JPEG0007702146000029.jpg103166

[0050] c. Second harmonic generation We can implement the same approach for calculating SHG in the cavity. However, in the case of SHG, the second harmonic mode must be extended to the eigenmodes of the cavity, while the pump input in the fundamental wave can be the built-in mode of the cavity. Ignoring the inverse conversion, the nonlinear dynamics for the SHG process can be described as follows:

Equation

[0051] JPEG0007702146000031.jpg53166

[0052] If there is only one coupling channel between the input and the cavity mode at the fundamental frequency, the cavity mode amplitude can be described as the input power as follows:

Equation

[0053] JPEG0007702146000033.jpg41166

[0054] JPEG0007702146000034.jpg42166

[0055] d. OPO of spherical dielectric particles The nonlinear coupling term in Equation 15 for the particles shown in FIGS. 3A - 3B when the pump is at the resonance frequency of the three magnetic modes is calculated as follows:

Equation

[0056] The modes are ordered as follows: ED, EQ, MD, and MQ. It can be seen that the off-diagonal terms can be even stronger than the diagonal terms. When ignoring the mode coupling (off-diagonal terms), the thresholds for these modes are, respectively, as follows: 3.99, 2783, 0.27, and 3.65 MW. However, due to the strong mode coupling that can be stronger than the diagonal terms based on Equation 33, the thresholds are reduced by a factor of 36 as shown in FIGS. 3A - 3B.

[0057] For the wavelength-scale OPO reported in FIGS. 6A - 6D, there are nine relevant eigenmodes. The resonant wavelengths of these modes are as follows: 2589, 1923, 1541, 1297, 3404, 2374, 1829, 1498, and 1273 nm. The first four modes are electric modes, and the last five modes are magnetic modes. They are sorted from the lowest order to the highest order. The Q-values of these modes are, respectively, 4, 19, 100, 520, 9, 37, 141, 600, and 2500. The non-linear coupling terms for pump excitation at 1110 nm are as follows:

Number

[0058] The non-linear coupling terms for pump excitation at 1125 nm are as follows:

Number

[0059] The eigenvalues at these two wavelengths are shown in FIG. 8. At the threshold, due to the large signal and idler frequency separation, it can be seen that the parametric gain is small. However, when a phase transition occurs from non-degenerate to degenerate cases, the gain increases rapidly.

[0060] JPEG0007702146000038.jpg37166

[0061] JPEG0007702146000039.jpg86166

[0062] By applying appropriate boundary conditions

[64] , this approach can be used to accurately find the quasi-normal modes of 3D resonators of arbitrary shape. In addition to the quasi-normal modes, this approach can find a series of background modes that depend on the boundary conditions and form a complete basis combined with the quasi-normal modes.

[0063] Due to the low Q of the background modes, their contribution to the OPO threshold is negligible. However, they can change the field distribution of the supermodes and their spectral responses beyond the threshold. The relationship between the quasi-normal modes and the density of states, ρ(ω), has been considered in previous studies [55, 42].

[0064] When having a series of states, the sum in Equation 22 is transformed into a formal form as follows:

Equation

[0065] Since the effect of the low Q background modes is negligible, Equation 35 can be discretized near the quasi-normal modes to simplify the numerical calculation:

Equation

[0066] 4. Practical Examples of OPO Figures 1A, 2B, 11C, 12A, and 13A - 13B illustrate an example of an OPO comprising a resonator 100 that includes a material 102 having a nonlinear susceptibility to generate an output electromagnetic field 104 in response to a pump electromagnetic field 106 input to the material. The output electromagnetic field has one or more output wavelengths that are longer than one or more pump wavelengths of the pump electromagnetic field. The resonator has dimensions 108 (e.g., at least one of diameter, width, length, or height) that are less than one or more output wavelengths in free space or on the order of one or more output wavelengths (within a factor of 2). Examples of dimensions include that the largest of dimensions 110 is less than 10 micrometers or that the resonator fits within a sphere or spherical volume having a radius of 5 microns, but are not limited thereto.

[0067] a. Particle examples Figures 1A and 2A illustrate an example in which the resonator comprises a particle 200 having dimensions 108, 110. Figure 4A illustrates an example in which the particle supports one or more quasi - normal electromagnetic modes 400 of the pump electromagnetic field. Figure 4D illustrates an example in which the particle supports one or more quasi - normal electromagnetic modes 402 of the output electromagnetic field. Figures 4A and 4D further illustrate quasi - normal electromagnetic modes comprising one or more multipole Mie resonances with the output electromagnetic field. In various embodiments, the particle (e.g., a nanoparticle or microparticle) has an arbitrary shape or cross - section. In one or more embodiments, the material comprises at least one of metal, dielectric, semiconductor, or polymer.

[0068] b. Plasmon resonator examples FIG. 11A shows an example comprising a plasmonic resonator 1100 in which the resonator supports one or more plasmon modes 1101 (see FIGS. 11C and 11D) of at least one of the pump electromagnetic field 106 or the output electromagnetic field 104. The resonator further comprises a structure 1102 having a gap 1104 that supports a plasmon mode overlapping the material 102. In one or more embodiments, the structure 1102 comprises a patterned metal layer 1106 disposed on the material 102. FIG. 11A further shows a patterned metal forming input coupler 1108 that couples the pump electromagnetic field to the resonator and an output coupler 1110 that couples the output electromagnetic field 104 outside the resonator. In one or more embodiments, the material comprises a dielectric, semiconductor, or polymer, and the structure defining the gap in the material comprises a patterned metal layer.

[0069] FIG. 11A further shows an example in which the resonator comprises an additional material 1112 having optical properties (e.g., relative permittivity) different from those of the material 102. In the example of FIG. 11a, the additional material 1112 is positioned over the gap 1104 and the material 102, although other configurations are possible. In some embodiments, the resonator has a structure 1102 that is adapted to or consists of a first interaction of the pump electromagnetic field and a second interaction of the output electromagnetic field, together with the additional material 1112 and the material 102, such that the efficiency of the parametric interaction between the pump electromagnetic field and the output electromagnetic field is increased compared to the case where the additional material 1112 is absent. In one or more embodiments, the parametric interaction is a non-linear interaction between the pump electromagnetic field and the output electromagnetic field, and the parametric interaction is provided by the material and / or the additional material. Examples of additional materials include, but are not limited to, polymers, glasses, linear materials, or materials having a refractive index less than two.

[0070] Table 1 compares the performance of wavelength-scale OPO plasmonic resonators having a valid oscillation threshold during operation.

Table 1

[0071] c. Dielectric resonator example FIG. 12A shows an example in which the resonator 100 includes a dielectric resonator 1200.

[0072] e. Inverse design example FIG. 13A shows an example of an inverse design OPO comprising a plurality of regions 1300 (e.g., pixels) having different relative permittivities and dimensions (e.g., thickness 1302) arranged to match the overlap of the pump electromagnetic field and the output electromagnetic field. In one embodiment, the resonator structure has a high Q value at the signal and pump wavelengths and is optimized to maximize the field overlap between the pump and the signal. In one embodiment, to optimize a wavelength-scale OPO, the structure is discretized into small pixels. Each pixel can be either a high-index / nonlinear material 1304 (blue, e.g., comprising a material 102 having a nonlinear susceptibility) or a low-index material 1306 (e.g., air). An optimization algorithm can be used to find the optimal configuration of pixels having a minimum OPO threshold. The structure can be in / out coupled, for example, from free space or through a waveguide.

[0073] f. OPO network example FIG. 14A shows an example of an OPO network 1400 comprising a plurality of OPOs in which the resonators are evanescently coupled, and FIG. 14B shows how the modes of the OPOs (output electromagnetic fields of the signal (s) and idler (i) in FIG. 14A) are coupled. In general, OPOs can be evanescently coupled (via an evanescent wave or field) or coupled through a coupling 1402. Examples of couplings include, but are not limited to, waveguides (e.g., waveguides between each pair of resonators) or auxiliary cavities (e.g., cavities containing two or more resonators). In some embodiments, each of the resonators is excited by a pump that outputs an electromagnetic field having a different phase and / or a different amplitude.

[0074] JPEG0007702146000043.jpg81166

[0075] In one or more embodiments, the sensor comprises an OPO network of FIG. 14A and one or more detectors 1404 coupled to the OPO to detect an output electromagnetic field, and senses at least one of a pump electromagnetic field or an environment near the network through detection of the output electromagnetic field by the detector.

[0076] In one or more embodiments, the coupling 1402 is adjusted to model an array of coupled spins such that the minimum threshold of the OPO network corresponds to the minimum energy configuration of the coupled spins in the array. Finding the minimum energy of a designed spin configuration can be applied to various optimization problems in biology, medicine, wireless communication, artificial intelligence and social networks. In one or more embodiments, operations are performed in an optical computer by using the coupling between OPOs.

[0077] 5. Process Steps Creation Method FIG. 15 is a flowchart showing a method of creating an optical parametric oscillator.

[0078] Block 1500 represents providing a resonator including a material having a nonlinear susceptibility to generate an output electromagnetic field in response to a pump electromagnetic field input to the material. The output electromagnetic field has one or more output wavelengths longer than one or more pump wavelengths of the pump electromagnetic field. The resonator has dimensions less than one or more output wavelengths, or on the order of one or more output wavelengths, in free space (e.g., air or the environment outside the material).

[0079] In one or more embodiments, the resonator is formed by using a lithography process and etching to remove a portion of the film.

[0080] In one or more embodiments, the resonator is designed using an inverse design process, where a plurality of regions having different relative permittivities and thicknesses are arranged to optimize or match the overlap of the pump electromagnetic field and the output electromagnetic field and / or to reduce the oscillation threshold of the OPO.

[0081] Block 1502 represents the end result, an OPO. The OPO can be implemented in many ways including, but not limited to, the following (see also FIGS. 1A, 2A, 4A - 4D, 11, 12A, 14A, 14B). 1. One or more optical parametric oscillators (OPOs), each comprising a resonator 100 comprising a material 102 having a nonlinear susceptibility that generates an output electromagnetic field 104 in response to a pump electromagnetic field 106 input to the material 102. The output electromagnetic field 104 has one or more output wavelengths that are longer than one or more pump wavelengths of the pump electromagnetic field. The resonator is a device having dimensions 108 that are less than one or more output wavelengths or on the order of one or more output wavelengths in free space. 2. The device of embodiment 1, wherein the resonator comprises a particle 200 having dimensions 110. 3. The device of embodiment 1 or 2, wherein the resonator supports one or more plasmon modes 1101 of at least one of the pump electromagnetic field or the output electromagnetic field. In one or more embodiments, the resonator supports a plasmon that confines the pump and / or output in the resonator. 4. The device of any of embodiments 1 or 3, wherein the resonator comprises a structure 1102 comprising a gap 1104 that supports a plasmon mode that overlaps the material. 5. The device of embodiment 4, further comprising an additional material 1112 having a second - order nonlinear susceptibility over the gap 1104. 6. The resonator comprises an additional substance 1112 having optical properties (e.g., relative permittivity) different from those of the substance. The resonator has the additional substance and the substance, and is adapted to the first interaction of the pump electromagnetic field and the second interaction of the output electromagnetic field, and has a structure (e.g., shape and / or dimensions) consisting of the first interaction of the pump electromagnetic field and the second interaction of the output electromagnetic field, so as to increase the efficiency of the parametric interaction between the pump electromagnetic field and the output electromagnetic field as compared with the case where there is no additional substance. The device according to any one of Examples 1 to 5. 7. The resonator comprises a polymer, glass, a linear substance, or at least one of refractive indices less than 2, or comprises an additional substance 1112 consisting essentially of a polymer, glass, a linear substance, or at least one of refractive indices less than 2. The device according to any one of Examples 1 to 6. In the example, the linear substance is defined as having no second-order susceptibility. In another example, the linear substance is defined as a substance that is "non-linear". 8. The additional substance 1112 comprises a polymer. The device according to any one of Examples 5 to 7. 9. The resonator is arranged to optimize the overlap between the pump electromagnetic field 106 and the output electromagnetic field 104, or to match the overlap between the pump electromagnetic field and the output electromagnetic field, and / or to reduce the oscillation threshold of the OPO, and comprises a plurality of regions or pixels 1300 having different relative permittivities and thicknesses 1302. The device according to any one of Examples 1 to 8. 10. The largest of the dimensions 110 is less than 10 microns, or the resonator fits within a sphere having a radius of 5 microns. The device according to any one of Examples 1 to 9. 11. The resonator supports one or more quasi-normal electromagnetic modes 400, 402 of the pump electromagnetic field 106 and / or quasi-normal electromagnetic modes of the output electromagnetic field 104. The device according to any one of Examples 1 to 10. 12. The quasi-normal electromagnetic mode comprises one or more multipole Mie resonances comprising the output electromagnetic field. The device of Example 11. 13. The device according to any one of Examples 1 to 12 further comprises a disk, a cylinder 1201 (FIG. 12a), or a sphere 201 (FIG. 2a) including the resonator. 14. The resonator is a device according to any one of Examples 1 to 13, having a polygonal cross-section or any cross-section (e.g., circular or non-uniform cross-section). 15. The resonator is a device according to any one of Examples 1 to 14, patterned by lithography. 16. The material 102 is a device according to any one of Examples 1 to 15, comprising at least one of a metal, a dielectric, a semiconductor, or a polymer. 17. The material 102 is a device according to any one of Examples 1 to 16, having at least one of a second susceptibility χ (2) or a third susceptibility χ (3) . 18. The device comprises a plurality of OPOs 1400, and the resonator is coupled evanescently 1402, or coupled through a waveguide or an auxiliary cavity 1402, according to any one of Examples 1 to 17. 19. The device comprises a plurality of OPOs that output a plurality of output electromagnetic fields 104 in response to a plurality of pump electromagnetic fields 106, each of the pump electromagnetic fields having at least one of a phase or amplitude different from the phase or amplitude of another pump electromagnetic field among the pump electromagnetic fields, according to any one of the Examples. 20. A sensor comprising a network 1400 comprising a plurality of OPOs according to any one of Examples 1 to 19, and one or more detectors 1404 coupled to detect the output electromagnetic field 104, and sensing the pump electromagnetic field or the environment near the network through the detection of the output electromagnetic field by the detector. 21. A network 1400 comprising a plurality of OPOs according to claim 1, and a coupling 1402 between the OPOs, the coupling 1402 being adjusted to model an array of coupled spins and such that the minimum threshold of each of the OPOs corresponds to one of the minimum energy configurations of the coupled spins in the array, an optical computer. 22. In one or more embodiments, the resonator is a structure having one or more optical properties and is shaped to support one or more resonances of the output electromagnetic field and / or the pump electromagnetic field. 23. Examples of the wavelengths of the pump electromagnetic field (e.g., having pump electromagnetic waves) and the output electromagnetic field (e.g., having pump electromagnetic waves) include, but are not limited to, wavelengths in the range from ultraviolet to mid-infrared. 24. In one or more embodiments, the output electromagnetic field comprises a signal (s) wave / field and an idler (i) wave / field.

[0082] Operation method FIG. 16 shows a method of operating an OPO including the following steps.

[0083] Block 1600 represents inputting a pump electromagnetic field into a resonator comprising a material having a nonlinear susceptibility that generates an output electromagnetic field in response to the pump electromagnetic field. As shown herein, the output electromagnetic field has one or more output wavelengths that are longer than one or more pump wavelengths of the pump electromagnetic field, and the resonator has dimensions less than one or more output wavelengths, or on the order of one or more output wavelengths, in free space.

[0084] Block 1602 represents configuring the OPO for at least one of the following: (1) An OPO operating in degeneracy such that at least one of the output wavelengths is at least twice one of the pump wavelengths (2) An output electromagnetic field comprising an optical frequency comb including a series of equally spaced frequency peaks (3) The output electromagnetic field has an output spectrum that is wider than the input spectrum of the pump electromagnetic field in frequency units measured at a level of 30 dB below the peak, or (4) A pump electromagnetic field including a continuous wave, time-varying, or pulsed electromagnetic field.

[0085] The OPO can be any of the OPOs of Examples 1-22 above.

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[0087] Conclusion Here, the description of the preferred embodiments of the present invention ends. The description of one or more embodiments of the present invention above has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise forms disclosed. Many modifications and variations are possible in light of the above teachings. It is intended that the scope of the present invention be defined not by this detailed description but by the claims appended hereto.

Claims

Claim 1 One or more optical parametric oscillators (OPOs), each comprising a resonator comprising a material having a non-linear susceptibility to generate an output electromagnetic field in response to a pump electromagnetic field input to the material, wherein the output electromagnetic field has one or more output wavelengths longer than one or more pump wavelengths of the pump electromagnetic field, the resonator has dimensions less than the one or more output wavelengths or on the order of the one or more output wavelengths in free space, the resonator provides resonance for the pump electromagnetic field and / or the output electromagnetic field, and a device in which multiple modes generated within the resonator interact non-linearly with each other through the pump electromagnetic field. Claim 2 The device of claim 1, wherein the resonator comprises particles having the dimensions. Claim 3 The device of claim 1, wherein the largest of the dimensions is less than 10 microns or the resonator fits within a sphere having a radius of 5 microns. Claim 4 The device of claim 1, wherein the resonator supports one or more quasi-normal electromagnetic modes of at least one of the pump electromagnetic field or the output electromagnetic field. Claim 5 The device of claim 4, wherein the quasi-normal electromagnetic mode comprises one or more multipole Mie resonances comprising the output electromagnetic field. Claim 6 The device of claim 4, further comprising a disk, cylinder, or sphere including the resonator. Claim 7 The device of claim 1, wherein the material comprises at least one of a metal, dielectric, semiconductor, or polymer. Claim 8 The device of claim 1, wherein the resonator supports one or more plasmon modes of at least one of the pump electromagnetic field or the output electromagnetic field. Claim 9 The device of claim 8, comprising a plurality of the OPOs, wherein the resonators are coupled evanescently or through a waveguide or auxiliary cavity. Claim 10 The device of claim 8, comprising a plurality of the OPOs that output a plurality of output electromagnetic fields in response to a plurality of the pump electromagnetic fields, wherein each of the pump electromagnetic fields can have at least one of the phase or amplitude different from the phase or amplitude of another pump electromagnetic field among the pump electromagnetic fields. Claim 11 The OPO of claim 1, wherein the resonator comprises a structure having a gap that supports a plasmon mode overlapping the material. Claim 12 ​ The OPO of claim 11, further comprising an additional substance having a second-order nonlinear susceptibility on top of the gap.

13. The resonator comprises an additional substance having a relative permittivity different from that of the substance, the resonator has the additional substance and the substance, and is adapted to the first interaction of the pump electromagnetic field and the second interaction of the output electromagnetic field, and has a structure consisting of the first interaction of the pump electromagnetic field and the second interaction of the output electromagnetic field, and increases the efficiency of the parametric interaction between the pump electromagnetic field and the output electromagnetic field as compared with the case where the additional substance is not present. The OPO of claim 1.

14. The resonator comprises a plurality of regions or pixels having different relative permittivities and thicknesses arranged to match the overlap between the pump electromagnetic field and the output electromagnetic field. The additional substance comprises at least one of a polymer, glass, a linear substance, or a refractive index less than 2. The OPO of claim 13.

15. A photonic integrated circuit comprising one or more of the resonators of claim 1.

16. The circuit of claim 15, further comprising a source of the pump electromagnetic field having free space located away from the circuit and coupled to the resonator, and not comprising a fiber optic coupling or waveguide for coupling the pump electromagnetic field applied from the free space.

17. A network comprising a plurality of the OPOs of claim 1, One or more detectors coupled to detect the output electromagnetic field and sensing at least one of the pump electromagnetic field or the environment near the network through detection of the output electromagnetic field by the detector. A sensor comprising.

18. A network comprising a plurality of the OPOs of claim 1, A coupling between the OPOs, adjusted to model an array of coupled spins, such that the minimum threshold of each of the OPOs corresponds to one minimum energy configuration of the coupled spins in the array. An optical computer comprising.

19. A method of operating an optical parametric oscillator (OPO), comprising: Inputting a pump electromagnetic field into a resonator comprising a substance having a nonlinear susceptibility that generates an output electromagnetic field in response to the pump electromagnetic field. The output electromagnetic field has one or more output wavelengths longer than one or more pump wavelengths of the pump electromagnetic field. The resonator has dimensions less than the one or more output wavelengths or on the order of the one or more output wavelengths in free space, The resonator provides resonance for the pump electromagnetic field and / or the output electromagnetic field, Multiple modes generated within the resonator interact nonlinearly with each other through the pump electromagnetic field, The OPO is configured for at least one of the following: The OPO operates at degeneracy such that at least one of the output wavelengths is twice at least one of the pump wavelengths, The output electromagnetic field comprises an optical frequency comb including a series of equidistant frequency peaks, The output electromagnetic field has an output spectrum wider than the input spectrum of the pump electromagnetic field in frequency units measured at a level of 30 dB below the peak, or The pump electromagnetic field comprises a continuous wave, time-varying, or pulsed electromagnetic field, a method. Claims 20 Comprising the step of providing a resonator comprising the material having a nonlinear susceptibility to generate an output electromagnetic field in response to a pump electromagnetic field input to the material, The output electromagnetic field has one or more output wavelengths longer than one or more pump wavelengths of the pump electromagnetic field, The resonator has dimensions less than the one or more output wavelengths or on the order of the one or more output wavelengths in free space, The resonator provides resonance for the pump electromagnetic field and / or the output electromagnetic field, A method of creating an optical parametric oscillator in which multiple modes generated within the resonator interact nonlinearly with each other through the pump electromagnetic field.

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