Abstract:
A method and system for optical systems based on parity-time symmetry and its breaking, and for nonreciprocal light transmission in a parity-time symmetric micro-resonator system are provided. The system includes an optical assembly that includes a first dissipative optical system and a second optical system coupled in energy transfer communication with the first optical system. The second optical system is configured to receive a continuous flow of energy from an external source and to transfer energy to the first optical system through the couple wherein the energy transferred to the first optical system from the second optical system is approximately equal to the energy dissipated in the first optical system, where the energy transferred to the first optical system from the second optical system is selectable using at least one of an amount of couple between the first optical system and the second optical system and a gain of the second optical system.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application claims priority to and the benefit of U.S. Provisional Patent Application Ser. No. 61/979,153 entitled, METHOD AND SYSTEM FOR PARITY-TIME SYMMETRIC OPTICS AND NONRECIPROCAL LIGHT TRANSMISSION, filed Apr. 14, 2014, which is incorporated by reference herein in its entirety. 
    
    
     Statement Regarding Federally Sponsored Research and Development: The U.S. Government has certain rights in this application as provided for by the terms of Army Research Office grant No. W911NF-12-1-0026. 
    
    
     BACKGROUND 
     This description relates to optical logical components, and, more particularly, to systems and methods of forming components using parity-time symmetry and it&#39;s breaking, exhibiting non-linearity-based nonreciprocal light transmission using parity-time symmetry. 
     A challenge in the field of photonics is developing new materials and devices with unconventional and advanced functionalities to control the flow of light on-chip. Recently, optical systems combining balanced loss and gain have been proposed to originate synthetic materials with properties that cannot be attained in materials having only loss or only gain. Such systems provide a platform to implement classical analogues of quantum systems described by non-Hermitian parity-time (PT) symmetric Hamiltonians and are expected to play a significant role in optics. PT synthetic materials utilize modulation of the coupling strength between individual molecular or macroscopic sub-components and sub-systems of a larger system, its refractive index, and gain and/or loss. This leads to phenomena such as asymmetric power oscillations between two waveguides (sub-systems of larger sub-systems) and unidirectional reflectionless light propagation. Parity-time symmetry can be obtained when using a pair of coupled systems (e.g., coupled waveguides and/or coupled resonators) with one dissipating light-waves while the other is amplifying an equal amount. The optical power in the gain system (i.e., active system, such as an active waveguide) can increase while optical power decays in the other. The propagation of light is non-reciprocal in that the power oscillation between the two coupled sub-systems is no longer symmetric. However, the transmission remains reciprocal. Loss-induced transparency, power oscillations violating left-right symmetry, PT-synthetic photonic lattices, and unidirectional invisibility have been demonstrated, but other phenomena such as nonreciprocal light transmission, prior the technology as disclosed herein; and coexisting coherent-perfect-absorption (CPA) and lasing are yet to be realized. These could benefit significantly from resonance structures exhibiting PT-symmetry. However, to date, experiments in PT-symmetric optics have been limited to waveguides in which resonances play no role. 
     BRIEF DESCRIPTION 
     In one aspect, an optical assembly includes a first dissipative optical system and a second optical system coupled in energy transfer communication with the first optical system. The second optical system is configured to receive a flow of energy, continuous or non-continuous/pulse, from an external source and to transfer energy to the first optical system through the coupling wherein the energy received by the second optical system is approximately equal to the energy dissipated in the first optical system, where the energy transferred to the first optical system from the second optical system is selectable using at least one of an amount of couple between the first optical system and the second optical system and a gain of the second optical system. An optical system can include, but is not limited to, any system that generates, emits, transmits, modulates, signal processes, switches, amplifies, detects or senses light, which can encompass systems having semi-conductor light emitters, lasers, optical waveguides, optical fibers, and micro-resonators. However, the technology as disclosed can benefit various resonance structures, beyond micro-resonators, including various resonance structures, such as Fabry-Perot cavities, plasmonic cavities, and resonators, and nano-micro-cavities and resonators, exhibiting parity-time symmetry. 
     As used herein, gain represents energy received by the second system from an external source and approximately equals to the loss (dissipation) of the first system. The first and second systems are configured such that they can exchange energy through the coupling. The coupling is the transfer or the means of transfer of energy from one element or medium to another. In the case of electronics, the coupling can be between circuit elements (for example via capacitive or inductive coupling); and in the case of optics, the coupling can be between optical devices or elements. A rate of energy exchange depends on the strength of coupling between the first and second systems. There can be coupling losses between a first system or element and a second system or element. In the case of optics, loss can be due a change in the index of refraction between first and second systems and a portion of the energy can be reflected back into the source component. Another reason for loss can be due to geometrical inconsistencies. The amount of energy transferred from the second system to the first system is controlled by the coupling strength and the amount of gain in the second system. 
     In another aspect, a method of nonreciprocal light transmission in a micro resonator system wherein the method includes coupling a first micro resonator optical system in energy communication to a second micro resonator optical system, operating the second micro resonator optical system in a dissipative mode wherein the second micro resonator optical system loses energy during the operation, adding gain to the first micro resonator optical system, transferring energy from the first micro resonator optical system to the second micro resonator optical system through the couple, and operating the first micro resonator optical system and second micro resonator optical system in a broken parity-time symmetry regime based on at least one of the amount of couple and the amount of gain added such that the micro resonator system exhibits nonreciprocal light transmission. 
     In yet another aspect, an optical logic component includes more than two resonant optical systems, at least some of the more than two resonant optical systems coupled together in energy transfer communication, a first portion of the more than two optical systems being dissipative systems, a second portion of the more than two optical systems are at least one of optically and electrically pumped to add a predetermined gain to associated ones of the first portion of the more than two resonant optical systems, said second portion configured to transfer energy to said first portion through the couple wherein the energy transferred to said first portion from said second portion is selectable to control a mode of operation of said optical logical component, where the energy transferred to said first portion from said second portion is selectable using at least one of an amount of couple between said first portion and said second portion and a gain of said second portion. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1-9  show example embodiments of the method and apparatus described herein. 
         FIG. 1  is a schematic block diagram of an optical system in accordance with an exemplary embodiment of the present disclosure. 
         FIG. 2  is a plan view of the system shown in  FIG. 1  illustrating the first resonator formed on a first chip and the second resonator shown in  FIG. 1  formed on a second chip. 
         FIG. 3  is a side view of the first resonator shown in  FIG. 1  and the second resonator shown in  FIG. 1 . 
         FIG. 4  is a graph of transmission spectra showing resonance lines of the first resonator shown in  FIG. 1  and the second resonator shown in  FIG. 1 . 
         FIG. 5  is a graph of transmission spectra showing resonance lines of the first resonator shown in  FIG. 1  and the second resonator shown in  FIG. 1 . 
         FIG. 6  is a graph showing a spectra of weak probe light. 
         FIGS. 7A-D  are graphs of real and imaginary parts of the eigenfrequencies calculated from the measured transmission spectra as a function of the coupling strength κ. 
         FIGS. 8A-C  are graphs illustrating an input-output relation in the PT-symmetric WGM resonators shown in  FIG. 1  showing a strong non-linear response in the broken-PT phase and a linear response in the unbroken-PT phase, and reciprocity in the linear regime of the broken-PT phase. 
         FIG. 9A  is a schematic block diagram of optical system with transmission in the forward direction from the first port shown in  FIG. 1  to the output port shown in  FIG. 1 . 
         FIG. 9B  illustrates that, in the forward direction as illustrated in  FIG. 9A , light can pass through the optical system, when both resonators are passive (no gain). 
         FIG. 9C  illustrates that, in the forward direction as illustrated in  FIG. 9A , light can pass through the system in the nonlinear regime, when the PT-symmetric resonators operate in the unbroken-symmetry region. 
         FIG. 9D  illustrates that, in the forward direction as illustrated in  FIG. 9A , light transmission is blocked in the nonlinear regime, when the PT-symmetric resonators operate in the broken-symmetry region. 
         FIG. 9E  is a schematic block diagram of the optical system shown in  FIG. 1  with transmission in the reverse direction from the output port shown in  FIG. 1  to the first port shown in  FIG. 1 . 
         FIG. 9F  illustrates that, in the backward direction as illustrated in  FIG. 9E , light can pass through the system, when both resonators are passive (no gain). 
         FIG. 9G  illustrates that, in the backward direction as illustrated in  FIG. 9E , light can pass through the system in the nonlinear regime, when the PT-symmetric resonators operate in the unbroken-symmetry region. 
         FIG. 9H  illustrates that, in the backward direction as illustrated in  FIG. 9E , light can pass through the system in the nonlinear regime, when the PT-symmetric resonators operate in broken-symmetry region. 
         FIG. 10  is a perspective view of an optical system in accordance with another example embodiment of the present disclosure. 
         FIG. 11  is a perspective view of an optical system in accordance with another example embodiment of the present disclosure. 
         FIG. 12  is a perspective view of an optical system in accordance with another example embodiment of the present disclosure. 
         FIG. 13  is a schematic block diagram of an optical system in accordance with another example embodiment of the present disclosure. 
         FIG. 14  is a schematic block diagram of an optical system in accordance with another example embodiment of the present disclosure. 
     
    
    
     Although specific features of various embodiments may be shown in some drawings and not in others, this is for convenience only. Any feature of any drawing may be referenced and/or claimed in combination with any feature of any other drawing. 
     Unless otherwise indicated, the drawings provided herein are meant to illustrate features of embodiments of the disclosure. These features are believed to be applicable in a wide variety of systems comprising one or more embodiments of the disclosure. As such, the drawings are not meant to include all conventional features known by those of ordinary skill in the art to be required for the practice of the embodiments disclosed herein. 
     DETAILED DESCRIPTION 
     The following detailed description illustrates embodiments of the disclosure by way of example and not by way of limitation. It is contemplated that the disclosure has general application to non-reciprocal transmission in physical systems including, but not limited to light wave, sound wave, heat wave, and electronic systems that are operated in a non-linear regime and more particularly to such systems operated in a parity-time (PT) symmetric configuration and operated in a broken PT symmetric phase such that the nonlinearity is enhanced in industrial, commercial, and residential applications. 
     Parity-time-symmetric quantum Hamiltonian systems have attracted increasing attention. It is shown that the eigenvalues of non-Hermitian Hamiltonians Ĥ†≠Ĥ can be entirely real if they respect PT-symmetry, PTĤ=ĤPT. PT-symmetric systems are open physical systems having balanced absorption (loss) and amplification (gain). Remarkably, such systems can exhibit a phase transition (spontaneous PT-symmetry breaking) if the parameter controlling the degree of non-Hermiticity exceeds a critical value. Beyond this threshold the eigenvalues become complex even though PTĤ=ĤPT still holds. PT symmetry has been studied experimentally and theoretically in various physical systems, with optics emerging as the most versatile platform to explore PT-symmetric applications. 
     In embodiments described herein, PT-symmetry breaking in a system of two directly-coupled on-chip whispering-gallery-mode resonators (WGMRs) is demonstrated. A transition to strong nonreciprocity in the PT symmetry-breaking phase is due to strong field-localization, which significantly enhances nonlinearity. In the linear regime, light transmission is still reciprocal regardless of whether the symmetry is broken or unbroken. Optical nonreciprocity has been a long sought-after goal in the study of PT-symmetry because of its substantial device implications for optical information processing. The embodiments described herein provide the first direct proof of nonreciprocity in PT-symmetric optics. A record-low power of 1 μW to observe nonlinearity-induced time-reversal-symmetry-breaking for nonreciprocal light transmission is achieved using a resonant structure. In one direction there is a complete absence of resonance peaks whereas in the other direction the transmission is resonantly enhanced. This feature is directly associated with the use of a resonant structure. As described herein, PT-symmetric microcavities are anticipated to be a starting point for unconventional optical systems utilizing resonance effects. For example, PT-symmetric microcavities are able to be used for building CPA-lasers and photonic analogues of topological insulators and for studying exceptional points in lasers. The embodiments described herein represent a significant advance towards a new generation of optical systems enabling on-chip manipulation and control of light propagation. 
     In a WGM resonator, light is confined by total internal reflection and circulates around the curved inner boundary of the resonator. WGM modes exhibit an evanescent tail that helps to couple light in and out of the resonator, and to probe the changes in or near the resonator for ultra-high performance sensing. Moreover, the evanescent tail makes it possible to couple directly two or more WGM resonators. The performance of a WGM resonator is determined by its quality factor Q, which represents the total loss (material, radiation, scattering and coupling losses) experienced by the light in the mode. 
     In one embodiment, the system can be composed of two directly coupled WGM microtoroidal resonators, each coupled to a different fiber-taper coupler. A first microtoroid includes a gain medium and may be referred to as an “active” resonator made from Er 3+ -doped silica formed using sol-gel synthesis and the second microtoroid is a “passive” (e.g., not populated with a gain medium) resonator made from silica without dopants. To have a controllable direct coupling between them, the microtoroids can be fabricated at the edges of two separate chips. The chips can be placed on nanopositioning to control precisely the distance and hence the coupling between the microtoroids. Optical gain in the first microtoroid can be provided by optically pumping the Er 3+  ions, which emit in the 1550 nm band, with a pump light at a 1460 nm band. The Q-factors of the first microtoroid and the second microtoroid in the 1550 nm wavelength band are 3.3×10 6  and 3×10 7 , respectively, and the first microtoroid has a Q-factor of 2.4×10 6  in the spectral band of the pump. 
     As the pump laser power is increased, the gain starts to compensate the losses of the first microtoroid in the 1550 nm band. This is reflected in the decreasing line width of the resonance line. When a weak probe light is coupled to first microtoroid together with the pump light, a strong resonance peak is observed, indicating that the weak probe signal has been amplified by the gain provided by Er 3+  ions. The resonance wavelength of the second microtoroid is thermally tuned through the thermo-optic effect of silica. By controlling the detuning in resonant wavelengths between first microtoroid and second microtoroid their coupling can be mediated in the 1550 nm band. In addition, the coupling between the resonators is modified by changing the distance between them. There is no coupling between the resonators in the 1460 nm band; thus, the pump light exists only in the first microtoroid. The ability to control the amplification and absorption ratio and the coupling strength in the exemplary embodiments described herein permits high versatility for investigating the novel physics of PT-symmetry. Previously, PT-symmetric photonic systems maintained a constant coupling strength, while the gain-to-loss ratio was varied. The embodiment described here allows to vary both the gain-to-loss ratio and the coupling strength. 
     The following description refers to the accompanying drawings, in which, in the absence of a contrary representation, the same numbers in different drawings represent similar elements. 
       FIG. 1  is a schematic block diagram of an optical system  100  in accordance with an example embodiment of the present disclosure. In the example embodiment, optical system  100  operates as an optical diode, in that light is transmitted in one direction through system  100  and not in the opposite direction through system  100 . In other embodiments, optical system  100  can be configured to operate as analogs to other logic and/or control elements typically found in electronics-based circuits. Optical system  100  includes a first whispering-gallery-mode (WGM) resonator  102  directly coupled to a second WGM resonator  104 , a first fiber-taper waveguide  106 , and a second fiber-taper waveguide  108 . Fiber-taper waveguides  106  and  108  each include a first normal portion  122 , a second normal portion  124 , and a tapered portion  126  between first normal portion  122  and second normal portion  124 . Tapered portion  126  has a diameter smaller than the wavelength of light transmitted by a light source introducing light into waveguides  106  and  108 . An evanescent field surrounds at least a part of tapered portion  126 . With WGM resonator  102  positioned proximate to tapered portion  126  (e.g., within the evanescent field), at least some light carried by waveguide  106  can be transmitted to or coupled into WGM resonator  102 . Similarly, light can be coupled out of or decoupled from WGM resonator  102  and coupled into tapered portion  126 . Light can be coupled between WGM resonators  102  and  104 . Additionally, light can be coupled into or out of WGM resonator  104  using waveguide  106 . 
     In exemplary embodiments, WGM resonator  102  is configured to receive light from tapered portion  126  and to allow the light to propagate within WGM resonator  102 . For example, a photon may travel around an ultra-high-Q WGM resonator  102  many times. Similarly, light can be coupled out of, or decoupled from, WGM resonator  102 . Light may be emitted by WGM resonator  102  proximate to tapered portion  126  of waveguide  108  and transmitted by second normal portion  124  away from system  100 . 
     A first resonator  102  can operate as an active resonator and a second resonator  104  can operate as a passive resonator. The first resonator  102  can be a microtoroid formed of Er 3+ -doped silica. The second resonator  104  can be a microtoroid formed of undoped silica. Optical system  100  can further includes a pump laser  110  configured to emit in the 1460 nanometer (nm) band to excite the Er 3+  ions to provide gain in the 1550 nm band. Light from pump laser  110  stays in first resonator  102  and does not couple to second resonator  104 . Light emitted from dopant couples from first resonator  102  to second resonator  104 . 
     In various embodiments, where first resonator  102  is formed of other dopants than Er 3+  ions, pump laser  110  is configured to emit in a different wavelength band corresponding to the excitation wavelength of the other dopant. Optical system  100  further includes a signal source  112  configured to emit a probe light signal in the 1550 nm band. 
     Optical system  100  can be used to demonstrate the broken and unbroken parity-time (PT) phases of operation as a function of coupling strength between first resonator  102  and second resonator  104 . For example, using only first fiber-taper waveguide  106  having a first port  114  and a second port  116 , pump laser  110  and the weaker signal source  112 , a probe laser can be input at first port  114  and the output transmission spectra can be monitored at second port  116  in the 1550 nm band. Without pump laser  110  operating, the coupled-resonator system  100  acts as a passive photonic molecule characterized by two super-modes whose spectral distance increases with increasing coupling strength by decreasing a distance between first resonator  102  and second resonator  104 . It should be noted that when gain being approximately the same as loss is maintained, the technology can transit from unbroken to broken region and vice versa by varying the coupling strength. Further, when the coupling strength is fixed, the technology can transit between broken and unbroken regions by varying the gain-to-loss ratio. The technology provides for both, whether varying the coupling strength or varying the gain-to-loss ratio. The specific example illustrated herein varies the coupling strength. Also, please note that gain and loss don&#39;t have to be exactly the same, but can be proximate, one with respect to the other. 
     System  100  becomes PT-symmetric when first resonator  102  is optically pumped to provide gain and second resonator  104  has a balanced loss. At a fixed gain-loss ratio (i.e., pump power), an output of second port  116  can be monitored as a function of the coupling strength κ, observing that there was a threshold coupling strength (κ PT ) at which a PT-symmetry phase transition occurred. For κ/κ PT &lt;1, system  100  is in a broken PT-symmetry phase, as seen in both a zero mode-splitting (shown in  FIG. 7A ) and a nonzero line width difference (shown in  FIG. 7B ). This indicates that the real parts of the eigenfrequencies have coalesced and that their imaginary parts are different. As κ/κ PT  approaches 1.0 from below, the line width difference decreases and frequencies bifurcate (mode-splitting). 
     In another embodiment, first resonator  102  can include a Q-factor of 2.0×10 7  and second resonator  104  can include a Q-factor of 3.0×10 7 . A power of pump laser  110  can be adjusted such that a balanced loss-gain ratio between first resonator  102  and second resonator  104  is approximately achieved. The transition from the broken to unbroken phase occurs at different coupling strengths for modes with different Q, that is, different initial loss (shown in  FIGS. 7D and 7D ). The lower the Q-factor, the higher the κ PT  for a PT phase transition. Typical transmission spectra in broken- and unbroken-PT-symmetric phases are shown, for example, in  FIGS. 8A, 8B, and 8C . 
     If the coupling between the resonators  102  and  104  is weak, the energy in active first resonator  102  cannot flow fast enough into passive second resonator  104  to compensate for the absorption or loss of resonator  104 . Therefore, system  100  cannot be in equilibrium and the eigenfrequencies are complex, implying exponential growth or decay. However, if the coupling strength exceeds a critical value, then system  100  can attain equilibrium because the energy in active first resonator  102  can flow rapidly enough into passive second resonator  104  to compensate the dissipation in resonator  104 . 
     The frequency bifurcation (splitting) is not in orthogonal directions (shown in  FIGS. 7A-D ) as would be expected for ideal systems with exactly balanced gain and loss. Instead, the bifurcation is smooth and the degree of smoothness (how much the system deviates from the exactly balanced case) depends on the power of pump laser  110  and thus on the gain-to-loss ratio. Equations of motion for coupled oscillators, show that, for unbalanced gain and loss, the eigenfrequencies are never exactly real. Instead, there is a region of κ where the difference in imaginary parts (line width difference) is large but the difference in real parts (mode splitting) is small (but nonzero). There is a second region where the line width difference is small but nonzero and the mode splitting is large. In practical implementations it is impossible to equalize the loss and gain exactly, so the mathematical prediction of a smooth bifurcation is physically realistic and is consistent with experimental observations (shown in  FIGS. 7A-D ). 
     PT-symmetric systems exhibit distinct behaviors including unidirectional or asymmetric transmission and invisibility, and enhanced or reduced reflections in the broken-PT-symmetric regime, where propagation is not invariant under the exchange of loss and gain. Nonreciprocal behavior in PT-symmetric system in the optical frequency range, which allows light to pass in only one direction is described using system  100 . The strong nonreciprocal light transmission is associated with nonlinearity enhancement due to strong field localization induced by PT-symmetry-breaking. In the linear regime, light transmission is still reciprocal. 
     A linear static dielectric system, even in the presence of gain and loss, cannot have a nonreciprocal response. However, a system with nonlinearity can exhibit very strong non-reciprocal as shown using system  100 . The transmission from first port  114  to an output port  118  is defined as forward transmission and transmission from output port  118  to first port  114  is defined as reverse transmission. The output spectra at first port  114  as the power of the input probe at output port  118  was varied when the system was in the broken- or unbroken-symmetry phases indicated clear nonlinear response in the symmetry-broken phase, in contrast to the linear response in the unbroken phase (shown in  FIG. 8A ). At low power levels, where the input-output relation was linear, the system was reciprocal in both the broken- and unbroken-symmetry regions ( FIGS. 8B and 8C ). This response shows a direct experimental clarification of the issue of reciprocity in PT symmetric systems; that is, PT symmetry or PT-symmetry breaking alone is not sufficient for nonreciprocal light transmission. 
     As demonstrated, as the input power is increased, system  100  remained in the linear regime for the unbroken-symmetry phase, whereas the input-output relation becomes nonlinear in the broken phase (shown in  FIG. 8A ). Such results indicate nonlinearity enhancement (i.e., the threshold for nonlinearity is lower) in the broken-symmetry phase, due to the stronger field localization into first resonator  102  with gain, as compared to the unbroken-symmetry phase. 
     Due to the stronger nonlinearity in the broken-symmetry case, the PT-symmetry transition is associated with a transition from reciprocal to nonreciprocal behavior. When pump laser  110  at first port  114  is OFF (first resonator  102  and second resonator  104  are passive) and a weak signal source  112  probe light is input at first port  114  or output port  118 , resonance peaks are observed in the forward or backward transmissions ( FIGS. 9B and 9F ) with no resolvable mode splitting. When pump laser  110  is set ON and the gain and loss are balanced as much as possible so as to operate the system in the unbroken-PT-symmetric region, transmission spectra showing amplified signals with clearly resolved split peaks are observed at the outputs in this strong-coupling region ( FIGS. 9C and 9G ). However, when the coupling strength is decreased so that the system transitions into the broken-symmetry region, forward transmission reduces to approximately zero ( FIG. 9D ) but the backward transmission remains high ( FIG. 9H ). The transmission spectra show a single resonance peak ( FIG. 9H ), as expected. Thus, in the broken-symmetry region, the input at output port  118  is transmitted to first port  114  at resonance; however the input at first port  114  cannot be transmitted to output port  118 , as opposed to what is observed for the unbroken-symmetry region. This indicates nonreciprocal light transport between first port  114  and output port  118 . 
     The advantages of the present design, which brings together PT-symmetric concepts with nonlinearity-induced nonreciprocal light transmission, over the non-PT schemes utilizing nonlinearity are a significant reduction in the input power to observe nonreciprocity (approximately ˜1 μW), higher contrast, small footprint and complete absence of the signal in one direction but resonantly enhanced transmission in the other direction. Similar nonreciprocity is observed between second port  116  and a third port  120 . These results imply that PT-symmetric WGMRs can have strong nonreciprocal effects (all-optical diode action) in the nonlinear regime with very low power threshold due to significant enhancement of nonlinearity in the broken-symmetry phase. 
       FIG. 2  is an image of a top of system  100  illustrating first resonator  102  formed on a first chip  202  and second resonator  104  formed on a second chip  204 . Chips  202  and  204  are positioned on respective nanopositioning systems to control precisely a distance between them and hence the coupling between first resonator  102  and second resonator  104 . 
       FIG. 3  is a side view of first resonator  102  and second resonator  104 . 
       FIG. 4  is a graph  400  of transmission spectra showing resonance lines of first resonator  102  and second resonator  104 . A first trace  402  shows a resonance line of second resonator  104  at approximately 1533.8 nm. A second trace  404  shows a resonance line of first resonator  102  at 1533.8 nm and a third trace  406  shows a resonance line of first resonator  102  at 1417.0 nm. 
       FIG. 5  is a graph  500  of transmission spectra showing resonance lines of first resonator  102 . A first trace  502  shows a resonance line of second resonator  104  at approximately 1533.8 nm without gain. A second trace  504  shows the same resonance line at 1533.8 nm with small gain and a third trace  506  shows the same resonance line at 1533.8 nm with medium amount of gain. Gain provided by Er 3+ -ions in first resonator  102  leads to narrower and deeper resonance lines as the power of pump laser  110  (gain) is increased, implying an increasing Q-factor. 
       FIG. 6  is a graph  600  showing a spectra of weak probe light. A trace  602  shows that weak probe light from signal source  112  is amplified when it is coupled to first resonator  102  together with light from pump laser  110 . An inset  604  shows a trace  606  of weak probe light that without the weak signal source  11  there is no resonance enhancement. 
       FIGS. 7A-D  are graphs of real and imaginary parts of the eigenfrequencies calculated from the measured transmission spectra as a function of the coupling strength κ. The real part of the eigenfrequencies correspond to the resonance frequencies thus mode splitting is the difference between the real part of the eigenfrequencies. Imaginary parts of the eigenfrequencies gives the line width information. Negative imaginary part of eigenfrequency corresponds to a lossy system, a positive imaginary part corresponds to gain. 
       FIGS. 7A and 7C  illustrate mode-splitting variation corresponding to the difference between the real parts and changes in the imaginary parts of the eigenfrequencies and  FIGS. 7B and 7D  illustrate and line width variation corresponding to the difference between the real parts and changes in the imaginary parts of the eigenfrequencies. Traces  702  and  704  illustrate when first and second resonators  102  and  104  are passive (no gain in first resonator  102 ) with Q factors 2.9×10 7  and 3.0×10 7 , respectively, for first resonator  102  and second resonator  104 . Traces  710 ,  712 ,  714 , and  716  illustrate when first resonator  102  is active and second resonator  104  is passive.  FIGS. 7C and 7D  illustrate an effect of the initial Q-factor (loss) of second resonator  104  on the eigenfrequencies. Two resonance modes with Q-factors 2.0×10 7    718  and  720 , and 3.0×10 7    722  and  724  are chosen for second resonator  104 . Shaded regions correspond to the broken-PT-symmetric region when gain and loss are balanced. 
       FIGS. 8A-C  are graphs illustrating an input-output relation in PT-symmetric WGM resonators and reciprocity in the linear regime.  FIG. 8A  includes a trace  802  illustrating a linear input-output relation in an unbroken-symmetry region. A trace  804  illustrates a nonlinear input-output relation in a broken-symmetry region. A signal applied at port  118  and transmitted to first port  114  is detected by a photodetector. Each data point represents an average of ten measurements, and an error bar represents a standard deviation. 
       FIG. 8B  illustrates transmission spectra in the linear regime (input power of ˜80 nW). A trace  806  illustrates reciprocal light transmission in a forward direction. A trace  808  illustrates reciprocal light transmission in a backward direction in the unbroken-symmetry region. The values represented in trace  806  were normalized with the signal detected at port  3  when the input was at port  4  and there was no coupling between the fiber tapers and the resonators. The values represented in trace  808  were normalized with the signal detected at port  2  when the input was at port  1  and there was no coupling between the fiber tapers and the resonators. 
       FIG. 8C  illustrates transmission spectra in the linear regime (input power of ˜80 nW). A trace  810  illustrates reciprocal light transmission in a forward direction. A trace  812  illustrates reciprocal light transmission in a backward direction in the unbroken-symmetry region. The values represented in trace  810  were normalized with the signal detected at port  3  when the input was at port  4  and there was no coupling between the fiber tapers and the resonators. The values represented in trace  812  were normalized with the signal detected at port  2  when the input was at port  1  and there was no coupling between the fiber tapers and the resonators. 
     The slight difference in the heights of the resonance peaks is attributed to the laser power fluctuations during frequency scanning and to the thermal fluctuations of the environment. These can be minimized, if not completely eliminated, by active stabilization and better control of the operational conditions. 
       FIG. 9A  is a schematic block diagram of optical system  100  with transmission in the forward direction from port  114  to port  118 . In the example embodiment, optical system  100  includes a first whispering-gallery-mode (WGM) resonator  102  directly coupled to a second WGM resonator  104 , a first fiber-taper waveguide  106 , and a second fiber-taper waveguide  108 .  FIGS. 9B-D  illustrate observed unidirectional transmission for PT-symmetric WGM microresonators in the nonlinear regime.  FIG. 9B  includes a trace  902  illustrating a transmission when both resonators are passive (no gain), the transmission is bi-directional (reciprocal), and light is transmitted in the forward direction (reverse direction shown in  FIG. 9F ).  FIG. 9C  includes a trace  904  illustrating a transmission in the unbroken-symmetry region, where the coupling exceeds the critical value and gain and loss are balanced and the transmission is still bi-directional (compare to  FIG. 9G ). Mode splitting due to coupling is resolved because gain compensates loss leading to narrower line widths.  FIG. 9D  includes a trace  906  illustrating that in the broken-symmetry region input in the forward direction does not reach the output (compare to  FIG. 9H ). 
       FIG. 9E  is a schematic block diagram of optical system  100  with transmission in the reverse direction from port  118  to port  114 .  FIGS. 9F-H  illustrate observed unidirectional transmission for PT-symmetric WGM microresonators in the nonlinear regime.  FIG. 9F  includes a trace  908  illustrating when both resonators are passive (no gain), the transmission is bi-directional (reciprocal), and light is transmitted in the reverse direction.  FIG. 9G  includes a trace  910  illustrating a transmission in the unbroken-symmetry region, where the coupling exceeds the critical value and gain and loss are balanced and the transmission is still bi-directional. Mode splitting due to coupling is resolved because gain compensates loss leading to narrower line widths.  FIG. 9H  includes a trace  912  illustrating that the broken-symmetry region transmission becomes unidirectional (non-reciprocal). Input in the reverse direction does reach the output. 
     This resembles the action of a diode and implies that an all-optical on-chip diode with PT-symmetric WGM microcavities operates in the broken-symmetry region. An inset  914  in  FIG. 9G  and an inset  916  in  FIG. 9H  shows the signal at port  114  when there is no input signal at port  118 . 
     In summary, using the methods described herein and illustrated in  FIGS. 9A-H  is an observed bidirectional transmission for passive photonic molecules (i.e. coupled resonators without gain), bidirectional light transmission in the unbroken symmetry region for PT-symmetric resonators, and unidirectional light transmission in the nonlinear and PT symmetry broken region. 
       FIGS. 10-14  represent various configurations of optical systems that include components configured to receive energy waves and to manipulate the energy waves to, for example, but not limited to generate logic outputs. In one embodiment, an optical diode is configured. In various embodiments, other forms of logic circuits are formed using sound wave devices to generate sound-based logic circuits or heat waves to generate heat-based logic circuits. The various configurations are formed using, for example, resonance chambers and controlling the coupling between the resonance chambers and the gain added to the system. 
       FIG. 10  is a perspective view of an optical system  1000  in accordance with another example embodiment of the present disclosure. Optical system  1000  includes a first whispering-gallery-mode (WGM) resonator  1002  closely directly coupled to a second WGM resonator  1004 , and a waveguide  1006 . First resonator  1002  operates as an active resonator and second resonator  1004  operates as a passive resonator. 
       FIG. 11  is a perspective view of an optical system  1100  in accordance with another example embodiment of the present disclosure. Optical system  1100  includes a first whispering-gallery-mode (WGM) resonator  1102  closely directly coupled to a second WGM resonator  1104 , and a waveguide  1106 . First resonator  1102  operates as an active resonator and second resonator  1104  operates as a passive resonator. 
       FIG. 12  is a perspective view of an optical system  1200  in accordance with another example embodiment of the present disclosure. Optical system  1200  includes a first whispering-gallery-mode (WGM) resonator  1202  closely directly coupled to a second WGM resonator  1204  and a third WGM resonator  1206 , and a waveguide  1208 . 
       FIG. 13  is a schematic block diagram of an optical system  1300  in accordance with another example embodiment of the present disclosure. Optical system  1300  includes a first whispering-gallery-mode (WGM) resonator  1302  loosely coupled to a second WGM resonator  1304 , a first waveguide  1306 , and a second waveguide  1308 . 
       FIG. 14  is a schematic block diagram of an optical system  1400  in accordance with another example embodiment of the present disclosure. Optical system  1400  includes a plurality of whispering-gallery-mode (WGM) resonators  1402 . WGM resonators  1402  may be loosely coupled to adjacent WGM resonators  1402 . Optical system  1400  includes a first waveguide  1404  and a second waveguide  1406 . 
     Each of first waveguide  1404  and second waveguide  1406  includes a tapered portion  1408  proximate each of WGM resonators  1402  for coupling tapered portion  1408  to a respective WGM resonator  1402 . Light received by WGM resonators  1402  from a respective waveguide  1404  or waveguide  1406  is confined within microtoroidal WGM resonators  1402 . For example, light may circulate through microtoroidal WGM resonator  1402  many times before being completely dissipated. Light coupled out of microtoroidal WGM resonator  1402  is received by tapered portion  1408  and carried away from system  1400  through one of first waveguide  1404  and a second waveguide  1406 . 
     The foregoing detailed description illustrates embodiments of the disclosure by way of example and not by way of limitation. It is contemplated that the disclosure has general application to the review and revision of advertisements. It is further contemplated that the methods and systems described herein may be incorporated into existing online advertising planning systems, in addition to being maintained as a separate stand-alone application. 
     While the disclosure has been described in terms of various specific embodiments, it will be recognized that the disclosure can be practiced with modification within the spirit and scope of the claims. 
     The above-described embodiments of a method and system of nonreciprocal light transmission in a chip scale system provides a cost-effective and reliable means for construction of optic analogs to electronic components. More specifically, the methods and systems described herein facilitate forming an optical assembly. 
     This written description uses examples to describe the disclosure, including the best mode, and also to enable any person skilled in the art to practice the disclosure, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the disclosure is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims.