Wireless energy transfer, including interference enhancement

[0001] Pursuant to U.S.C. §119(e), this application claims priority to U.S. Provisional Application Ser. No. 61/127,661, filed May 14, 2008.

[0002] This application is also related by subject matter to the following commonly owned applications: U.S. Utility patent application Ser. No. 12/055,963, filed Mar. 26, 2008; U.S. Utility patent application Ser. No. 11/481,077, filed Jul. 5, 2006; U.S. Provisional Application Ser. No. 60/698,442, filed Jul. 12, 2005; U.S. Provisional Application Ser. No. 60/908,383, filed Mar. 27, 2007; U.S. Provisional Application Ser. No. 60/908,666, filed Mar. 28, 2007; and International Application No. PCT/US2007/070892, filed Jun. 11, 2007.

[0003] The contents of the prior applications are incorporated herein by reference in their entirety.

[0004] This invention was made with government support under grant number W911NF-07-D-0004 awarded by the Army Research Office. The government has certain rights in this invention.

[0005] The disclosure relates to wireless energy transfer. Wireless energy transfer can for example, be useful in such applications as providing power to autonomous electrical or electronic devices.

[0006] Radiative modes of omni-directional antennas (which work very well for information transfer) are not suitable for such energy transfer, because a vast majority of energy is wasted into free space. Directed radiation modes, using lasers or highly-directional antennas, can be efficiently used for energy transfer, even for long distances (transfer distance L_TRANS>>L_DEV, where L_DEV is the characteristic size of the device and/or the source), but require existence of an uninterruptible line-of-sight and a complicated tracking system in the case of mobile objects. Some transfer schemes rely on induction, but are typically restricted to very close-range (L_TRANS<<L_DEV) or low power (˜mW) energy transfers.

[0007] The rapid development of autonomous electronics of recent years (e.g. laptops, cell-phones, house-hold robots, that all typically rely on chemical energy storage) has led to an increased need for wireless energy transfer.

[0008] Efficient wireless energy-transfer between two resonant objects can be achieved at mid-range distances, provided these resonant objects are designed to operate in the ‘strong-coupling’ regime. We describe an implementation of a method to increase the efficiency of energy-transfer or to suppress the power radiated, which can be harmful or a cause of interference to other communication systems, by utilizing destructive interference between the radiated far-fields of the resonant coupled objects. ‘Strong coupling’ is a necessary condition for efficient energy-transfer, in the absence of far-field interference. ‘Strong coupling’ can be demonstrated in the case of realistic systems: self-resonant conducting coils, capacitively-loaded conducting coils, inductively-loaded conducting rods and dielectric disks, all bearing high-Q electromagnetic resonant modes. Also, an analytical model can be developed to take far-field interference into account for wireless energy-transfer systems. The analytical model can be used to demonstrate the efficiency enhancement and radiation suppression, in the presence of interference. In an example implementation, we describe improved performance based on the above principles in the case of two realistic systems: capacitively-loaded conducting coils and dielectric disks, both bearing high-Q electromagnetic resonant modes and far-field interference.

[0009] In an aspect, an apparatus for use in wireless energy transfer includes a first resonator structure configured for energy transfer with a second resonator structure, over a distance D larger than a characteristic size L₁ of said first resonator structure and larger than a characteristic size L₂ of said second resonator structure. The energy transfer has a rate κ and is mediated by evanescent-tail coupling of a resonant field of the first resonator structure and a resonant field of the second resonator structure. The resonant field of the first resonator structure has a resonance angular frequency ω₁, a resonance frequency-width Γ₁, and a resonance quality factor Q₁=ω₁/2Γ₁, and the said resonant field of the second resonator structure has a resonance angular frequency ω₂, a resonance frequency-width Γ₂ , and a resonance quality factor Q₂=ω₂/2Γ₂. The absolute value of the difference of said angular frequencies ω₁ and ω₂ is smaller than the broader of said resonant widths Γ₁ and Γ₂. The apparatus also includes a power supply coupled to the first structure and configured to drive the first resonator structure or the second resonator structure at an angular frequency away from the resonance angular frequencies and shifted towards a frequency corresponding to an odd normal mode for the resonator structures to reduce radiation from the resonator structures by destructive far-field interference.

[0010] In some examples, the power supply is configured to drive the first resonator structure or the second resonator structure at the angular frequency away from the resonance angular frequencies and shifted towards the frequency corresponding to an odd normal mode for the resonator structures to substantially suppress radiation from the resonator structures by destructive far-field interference.

[0011] In an aspect, a method for wireless energy transfer involves a first resonator structure configured for energy transfer with a second resonator structure, over a distance D larger than a characteristic size L₁ of said first resonator structure and larger than a characteristic size L₂ of said second resonator structure, wherein the energy transfer has a rate κ and is mediated by evanescent-tail coupling of a resonant field of the first resonator structure and a resonant field of the second resonator structure, said resonant field of the first resonator structure has a resonance angular frequency ω₁, a resonance frequency-width Γ₁, and a resonance quality factor Q₁=ω₁/2Γ₁, and said resonant field of the second resonator structure has a resonance angular frequency ω₂, a resonance frequency-width Γ₂, and a resonance quality factor Q₂=ω₂/2Γ₂, the absolute value of the difference of said angular frequencies ω₁ and ω₂ is smaller than the broader of said resonant widths Γ₁ and Γ₂. The method includes driving the first resonator structure or the second resonator structure at an angular frequency away from the resonance angular frequencies and shifted towards a frequency corresponding to an odd normal mode for the resonator structures to reduce radiation from the resonator structures by destructive far-field interference.

[0012] In some examples, the first resonator structure or the second resonator structure is driven at the angular frequency away from the resonance angular frequencies and shifted towards the frequency corresponding to an odd normal mode for the resonator structures to substantially suppress radiation from the resonator structures by destructive far-field interference.

[0013] In an aspect, an apparatus for use in wireless energy transfer includes a first resonator structure configured for energy transfer with a second resonator structure, over a distance D larger than a characteristic size L₁ of said first resonator structure and larger than a characteristic size L₂ of said second resonator structure. The energy transfer has a rate κ and is mediated by evanescent-tail coupling of a resonant field of the first resonator structure and a resonant field of the second resonator structure. The resonant field of the first resonator structure has a resonance angular frequency ω₁, a resonance frequency-width Γ₁, and a resonance quality factor Q₁=ω₁/2Γ₁, and the resonant field of the second resonator structure has a resonance angular frequency ω₂, a resonance frequency-width Γ₂, and a resonance quality factor Q₂=ω₂/2Γ₂. The absolute value of the difference of said angular frequencies ω₁ and ω₂ is smaller than the broader of said resonant widths Γ₁ and Γ₂. For a desired range of the distances D, the resonance angular frequencies for the resonator structures increase transmission efficiency T by accounting for radiative interference, wherein the increase is relative to a transmission efficiency T calculated without accounting for the radiative interference.

[0014] In some examples, the resonance angular frequencies for the resonator structures are selected by optimizing the transmission efficiency T to account for both a resonance quality factor U and an interference factor V.

[0015] In an aspect, a method involves designing a wireless energy transfer apparatus, the apparatus including a first resonator structure configured for energy transfer with a second resonator structure, over a distance D larger than a characteristic size L₁ of said first resonator structure and larger than a characteristic size L₂ of said second resonator structure, wherein the energy transfer has a rate κ and is mediated by evanescent-tail coupling of a resonant field of the first resonator structure and a resonant field of the second resonator structure, wherein said resonant field of the first resonator structure has a resonance angular frequency ω₁, a resonance frequency-width Γ₁, and a resonance quality factor Q₁=ω₁/2Γ₁, and said resonant field of the second resonator structure has a resonance angular frequency ω₂, a resonance frequency-width Γ₂ and a resonance quality factor Q₂=ω₂/2Γ₂, wherein the absolute value of the difference of said angular frequencies ω₁ and ω₂ is smaller than the broader of said resonant widths Γ₁ and Γ₂. The method includes selecting the resonance angular frequencies for the resonator structures to substantially optimize the transmission efficiency by accounting for radiative interference between the resonator structures.

[0016] In some examples, the resonance angular frequencies for the resonator structures are selected by optimizing the transmission efficiency T to account for both a resonance quality factor U and an interference factor V.

[0017] In an aspect, an apparatus for use in wireless energy transfer includes a first resonator structure configured for energy transfer with a second resonator structure over a distance D. The energy transfer is mediated by evanescent-tail coupling of a resonant field of the first resonator structure and a resonant field of the second resonator structure, with a coupling factor k. The resonant field of the first resonator structure has a resonance angular frequency ω₁, a resonance frequency-width Γ₁, and a resonance quality factor Q₁=ω₁/2Γ₁, and is radiative in the far field, with an associated radiation quality factor Q_1,rad≧Q₁, and the resonant field of the second resonator structure has a resonance angular frequency ω₂, a resonance frequency-width Γ₂, and a resonance quality factor Q₂=ω₂/2Γ₂, and is radiative in the far field, with an associated radiation quality factor Q_2,rad≧Q₂. An absolute value of a difference of said angular frequencies ω₁ and ω₂ is smaller than broader of said resonant widths Γ₁ and Γ₂, and an average resonant angular frequency is defined as ω_o=√{square root over (ω₁ω₂)}, corresponding to an average resonant wavelength λ_o=2πc/ω_o, where c is the speed of light in free space, and a strong-coupling factor being defined as U=k√{square root over (Q₁Q₂)}. The apparatus is configured to employ interference between said radiative far fields of the resonant fields of the first and second resonator, with an interference factor V_rad, to reduce a total amount of radiation from the apparatus compared to an amount of radiation from the apparatus in the absence of interference, a strong-interference factor being defined as V=V_rad√{square root over ((Q₁/Q_1,rad)(Q₂/Q_2,rad))}{square root over ((Q₁/Q_1,rad)(Q₂/Q_2,rad))}.

[0018] The following are examples within the scope of this aspect.

[0019] The apparatus has Q₁/Q_1,rad≧0.01 and Q₂/Q_2,rad≧0.01. The apparatus has Q₁/Q_1,rad≧0.1 and Q₂/Q_2,rad≧0.1. The apparatus has D/λ_o larger than 0.001 and the strong-interference factor V is larger than 0.01. The apparatus has D/λ_o larger than 0.001 and the strong-interference factor V is larger than 0.1. The apparatus includes the second resonator structure.

[0020] During operation, a power generator is coupled to one of the first and second resonant structure, with a coupling rate κ_g, and is configured to drive the resonator structure, to which it is coupled, at a driving frequency f, corresponding to a driving angular frequency ω=2πf, wherein U_g is defined as κ_g/Γ₁, if the power generator is coupled to the first resonator structure and defined as κ_g/Γ₂, if the power generator is coupled to the second resonator structure. The driving frequency is different from the resonance frequencies of the first and second resonator structures and is closer to a frequency corresponding to an odd normal mode of the system of the two resonator structures, wherein the detuning of the first resonator from the driving frequency is defined as D₁=(ω−ω₁)/Γ₁ and the detuning of the second resonator structure from the driving frequency is defined as D₂=(ω−ω₂)/Γ₂.

[0021] D₁ is approximately equal to UV_rad and D₂ is approximately equal to UV_rad. U_g is chosen to maximize the ratio of the energy-transfer efficiency to the radiation efficiency. U_g is approximately equal to √{square root over (1+U²−V_rad²U²+V²−2VV_rad)}. f is at least larger than 100 kHz and smaller than 500 MHz. f is at least larger than 1 MHz and smaller than 50 MHz. The apparatus further includes the power generator. During operation, a power load is coupled to the resonant structure to which the power generator is not coupled, with a coupling rate κ₁, and is configured to receive from the resonator structure, to which it is coupled, a usable power, wherein U₁ is defined as κ₁/Γ₁, if the power load is coupled to the first resonator structure and defined as κ₁/Γ₂, if the power load is coupled to the second resonator structure. U₁ is chosen to maximize the ratio of the energy-transfer efficiency to the radiation efficiency. The driving frequency is different from the resonance frequencies of the first and second resonator structures and is closer to a frequency corresponding to an odd normal mode of the system of the two resonator structures, wherein the detuning of the first resonator from the driving frequency is defined as D₁=(ω−ω₁)/Γ₁ and is approximately equal to UV_rad, and the detuning of the second resonator structure from the driving frequency is defined as D₂=(ω−ω₂)/Γ₂ and is approximately equal to UV_rad, and U₁ is approximately equal to √{square root over (1+U²−V_rad²+U²+V²−VV_rad)}.

[0022] At least one of the first and second resonator structures comprises a capacitively loaded loop or coil of at least one of a conducting wire, a conducting Litz wire, and a conducting ribbon. The characteristic size of said loop or coil is less than 30 cm and the width of said conducting wire or Litz wire or ribbon is less than 2 cm. The characteristic size of said loop or coil is less than 1 m and the width of said conducting wire or Litz wire or ribbon is less than 2 cm.

[0023] The apparatus further includes a feedback mechanism for maintaining the resonant frequency of one or more of the resonant objects. The feedback mechanism comprises an oscillator with a fixed driving frequency and is configured to adjust the resonant frequency of the one or more resonant objects to be detuned by a fixed amount with respect to the fixed frequency.

[0024] In an aspect, an apparatus for use in wireless energy transfer includes a first resonator structure configured for energy transfer with a second resonator structure over a distance D. The energy transfer is mediated by evanescent-tail coupling of a resonant field of the first resonator structure and a resonant field of the second resonator structure, with a coupling factor k. The resonant field of the first resonator structure has a resonance angular frequency ω₁, a resonance frequency-width Γ₁, and a resonance quality factor Q₁=ω₁/2Γ₁, and is radiative in the far field, with an associated radiation quality factor Q_1,rad≧Q₁, and the resonant field of the second resonator structure has a resonance angular frequency ω₂, a resonance frequency-width Γ₂, and a resonance quality factor Q₂=ω₂/2Γ₂, and is radiative in the far field, with an associated radiation quality factor Q_2,rad≧Q₂. An absolute value of a difference of said angular frequencies ω₁ and ω₂ is smaller than the broader of said resonant widths Γ₁ and Γ₂, and an average resonant angular frequency is defined as ω_o=√{square root over (ω₁ω₂)}, corresponding to an average resonant wavelength λ_o=2πc/ω_o, where c is the speed of light in free space, and a strong-coupling factor is defined as U=k√{square root over (Q₁Q₂)}. The apparatus is configured to employ interference between said radiative far fields of the resonant fields of the first and second resonator, with an interference factor V_rad, to increase efficiency of energy transfer for the apparatus compared to efficiency for the apparatus in the absence of interference, the strong-interference factor being defined as V=V_rad√{square root over ((Q₁/Q_1,rad)(Q₂/Q_2,rad))}{square root over ((Q₁/Q_1,rad)(Q₂/Q_2,rad))}.

[0025] The following are examples within the scope of this aspect.

[0026] The apparatus has Q₁/Q_1,rad≧0.05 and Q₂/Q_2,rad≧0.05. The apparatus has Q₁/Q_1,rad≧0.5 and Q₂/Q_2,rad≧0.5. The apparatus has D/λ_o larger than 0.01 and the strong-interference factor V is larger than 0.05. The apparatus has D/λ₁ larger than 0.01 and the strong-interference factor V is larger than 0.5. The apparatus further includes the second resonator structure.

[0027] During operation, a power generator is coupled to one of the first and second resonant structure, with a coupling rate κ_g, and is configured to drive the resonator structure, to which it is coupled, at a driving frequency f, corresponding to a driving angular frequency ω=2πf, wherein U_g is defined as κ_g/Γ₁, if the power generator is coupled to the first resonator structure and defined as κ_g/Γ₂, if the power generator is coupled to the second resonator structure. The driving frequency is different from the resonance frequencies of the first and second resonator structures and is closer to a frequency corresponding to an odd normal mode of the system of the two resonator structures, wherein the detuning of the first resonator from the driving frequency is defined as D₁=(ω−ω₁)/Γ₁ and the detuning of the second resonator structure from the driving frequency is defined as D₂=(ω−ω₂)/Γ₂.

[0028] D₁ is approximately equal to UV and D₂ is approximately equal to UV. U_g is chosen to maximize the energy-transfer efficiency. U_g is approximately equal to √{square root over ((1+U²)(1−V²))}{square root over ((1+U²)(1−V²))}. f is at least larger than 100 kHz and smaller than 500 MHz. f is at least larger than 1 MHz and smaller than 50 MHz. The apparatus further includes the power generator.

[0029] During operation, a power load is coupled to the resonant structure to which the power generator is not coupled, with a coupling rate κ₁, and is configured to receive from the resonator structure, to which it is coupled, a usable power, wherein U₁ is defined as κ₁/Γ₁, if the power load is coupled to the first resonator structure and defined as κ₁/Γ₂, if the power load is coupled to the second resonator structure. U₁ is chosen to maximize the energy-transfer efficiency. The driving frequency is different from the resonance frequencies of the first and second resonator structures and is closer to a frequency corresponding to an odd normal mode of the system of the two resonator structures, wherein the detuning of the first resonator from the driving frequency is defined as D₁=(ω−ω₁)/Γ₁ and is approximately equal to UV, and the detuning of the second resonator structure from the driving frequency is defined as D₂=(ω−ω₂)/Γ₂ and is approximately equal to UV, and U₁ is approximately equal to √{square root over ((1+U²)(1−V²))}{square root over ((1+U²)(1−V²))}.

[0030] At least one of the first and second resonator structures comprises a capacitively loaded loop or coil of at least one of a conducting wire, a conducting Litz wire, and a conducting ribbon. The characteristic size of said loop or coil is less than 30 cm and the width of said conducting wire or Litz wire or ribbon is less than 2 cm. The characteristic size of said loop or coil is less than 1 m and the width of said conducting wire or Litz wire or ribbon is less than 2cm. The apparatus includes a feedback mechanism for maintaining the resonant frequency of one or more of the resonant objects. The feedback mechanism comprises an oscillator with a fixed driving frequency and is configured to adjust the resonant frequency of the one or more resonant objects to be detuned by a fixed amount with respect to the fixed frequency. The feedback mechanism is configured to monitor an efficiency of the energy transfer, and adjust the resonant frequency of the one or more resonant objects to maximize the efficiency. The resonance angular frequencies for the resonator structures are selected to optimize the energy-transfer efficiency by accounting for both the strong-coupling factor U and the strong-interference interference factor V.

[0031] In an aspect, a method for wireless energy transfer includes providing a first resonator structure configured for energy transfer with a second resonator structure over a distance D, wherein the energy transfer is mediated by evanescent-tail coupling of a resonant field of the first resonator structure and a resonant field of the second resonator structure, with a coupling factor k, wherein said resonant field of the first resonator structure has a resonance angular frequency ω₁, a resonance frequency-width Γ₁, and a resonance quality factor Q₁=ω₁/2Γ₁, and is radiative in the far field, with an associated radiation quality factor Q_1,rad≧Q₁, and resonant field of the second resonator structure has a resonance angular frequency ω₂, a resonance frequency-width Γ₂, and a resonance quality factor Q₂=ω₂/2Γ₂, and is radiative in the far field, with an associated radiation quality factor Q_2,rad≧Q₂, wherein an absolute value of a difference of said angular frequencies ω₁ and ω₂ is smaller than broader of said resonant widths Γ₁ and Γ₂, and an average resonant angular frequency is defined as ω_o=√{square root over (ω₁ω₂)}, corresponding to an average resonant wavelength λ_o=2πc/ω_o, where c is the speed of light in free space, and the strong-coupling factor is defined as U=k√{square root over (Q₁Q₂)}, and employing interference between said radiative far fields of the resonant fields of the first and second resonator, with an interference factor V_rad, to reduce a total amount of radiation from the first and second resonator compared to an amount of radiation from the first and second resonator in the absence of interference, a strong-interference factor being defined as V=V_rad√{square root over ((Q₁/Q_1,rad)(Q₂/Q_2,rad))}{square root over ((Q₁/Q_1,rad)(Q₂/Q_2,rad))}.

[0032] The following are examples within the scope of this aspect.

[0033] The method has Q₁/Q_1,rad≧0.01 and Q₂/Q_2,rad≧0.01. During operation, a power generator is coupled to one of the first and second resonant structure and is configured to drive the resonator structure, to which it is coupled, at a driving frequency f, corresponding to a driving, angular frequency ω=2πf, wherein the driving frequency is different from the resonance frequencies of the first and second resonator structures and is closer to a frequency corresponding to an odd normal mode of the system of the two resonator structures. During operation, a power load is coupled to the resonant structure to which the power generator is not coupled and is configured to receive from the resonator structure, to which it is coupled, a usable power. In an aspect, a method for wireless energy transfer includes providing a first resonator structure configured for energy transfer with a second resonator structure over a distance D, wherein the energy transfer is mediated by evanescent-tail coupling of a resonant field of the first resonator structure and a resonant field of the second resonator structure, with a coupling factor k, wherein said resonant field of the first resonator structure has a resonance angular frequency ω₁, a resonance frequency-width Γ₁, and a resonance quality factor Q₁=ω₁/2Γ₁, and is radiative in the far field, with an associated radiation quality factor Q_1,rad≧Q₁, and said resonant field of the second resonator structure has a resonance angular frequency ω₂, a resonance frequency-width Γ₂, and a resonance quality factor Q₂=ω₂/2Γ₂, and is radiative in the far field, with an associated radiation quality factor Q_2,rad≧Q₂, wherein an absolute value of the difference of said angular frequencies ω₁ and ω₂ is smaller than the broader of said resonant widths Γ₁ and Γ₂, and an average resonant angular frequency is defined as ω_o=√{square root over (ω₁ω₂)}, corresponding to an average resonant wavelength λ_o=2πc/ω_o, where c is the speed of light in free space, and the strong-coupling factor is defined as U=k√{square root over (Q₁Q₂)}, and employing interference between said radiative far fields of the resonant fields of the first and second resonator, with an interference factor V_rad, to increase efficiency of energy transfer between the first and second resonator compared to efficiency of energy transfer between the first and second resonator in the absence of interference, a strong-interference factor being defined as V=V_rad√{square root over ((Q₁/Q_1,rad)(Q₂/Q_2,rad))}{square root over ((Q₁/Q_1,rad)(Q₂/Q_2,rad))}.

[0034] The following are examples within the scope of this aspect.

[0035] The method has Q₁/Q_1,rad≧0.05 and Q₂/Q_2,rad≧0.05. During operation, a power generator is coupled to one of the first and second resonant structure and is configured to drive the resonator structure, to which it is coupled, at a driving frequency f, corresponding to a driving angular frequency ω=2πf, wherein the driving frequency is different from the resonance frequencies of the first and second resonator structures and is closer to a frequency corresponding to an odd normal mode of the system of the two resonator structures. During operation, a power load is coupled to the resonant structure to which the power generator is not coupled and is configured to receive from the resonator structure, to which it is coupled, a usable power. The resonance angular frequencies for the resonator structures are selected to optimize the energy-transfer efficiency by accounting for both the strong-coupling factor U and the strong-interference interference factor V.

[0036] Various examples may include any of the above features, alone or in combination. Other features, objects, and advantages of the disclosure will be apparent from the following detailed description.

[0037] For example, in some embodiments Q₁>100 and Q₂>100, Q₁>300 and Q₂>300, Q₁>500 and Q₂>500, or Q₁>1000 and Q₂>1000. In some embodiments, Q₁>100 or Q₂>100, Q₁>300 or Q₂>300, Q₁>500 or Q₂>500, or Q₁>1000 or Q₂>1000. Furthermore, for example, in some embodiments, the coupling to loss ratio

[0038] $\frac{\kappa }{\sqrt{{\Gamma }_1{\Gamma }_2}}>0.5,\frac{\kappa }{\sqrt{{\Gamma }_1{\Gamma }_2}}>1,\frac{\kappa }{\sqrt{{\Gamma }_1{\Gamma }_2}}>2,\mathrm{or}\frac{\kappa }{\sqrt{{\Gamma }_1{\Gamma }_2}}>5.$

1. Efficient Energy-Transfer by ‘Strongly Coupled’ Resonances

[0072] FIG. 1 shows a schematic that generally describes one example of the invention, in which energy is transferred wirelessly between two resonant objects. Referring to FIG. 1, energy is transferred, over a distance D, between a resonant source object having a characteristic size r₁ and a resonant device object of characteristic size r₂. Both objects are resonant objects. The wireless non-radiative energy transfer is performed using the field (e.g. the electromagnetic field or acoustic field) of the system of two resonant objects.

[0073] The characteristic size of an object can be regarded as being equal to the radius of the smallest sphere which can fit around the entire object. The characteristic thickness of an object can be regarded as being, when placed on a flat surface in any arbitrary configuration, the smallest possible height of the highest point of the object above a flat surface. The characteristic width of an object can be regarded as being the radius of the smallest possible circle that the object can pass through while traveling in a straight line. For example, the characteristic width of a cylindrical object is the radius of the cylinder.

[0074] It is to be understood that while two resonant objects are shown in the example of FIG. 1, and in many of the examples below, other examples can feature three or more resonant objects. For example, in some examples, a single source object can transfer energy to multiple device objects. In some examples, energy can be transferred from a first resonant object to a second resonant object, and then from the second resonant object to a third resonant object, and so forth.

[0075] Initially, we present a theoretical framework for understanding non-radiative wireless energy transfer. Note however that it is to be understood that the scope of the invention is not bound by theory.

[0076] Different temporal schemes can be employed, depending on the application, to transfer energy between two resonant objects. Here we will consider two particularly simple but important schemes: a one-time finite-amount energy-transfer scheme and a continuous finite-rate energy-transfer (power) scheme.

1.1 Finite-Amount Energy-Transfer Efficiency

[0077] Let the source and device objects be 1, 2 respectively and their resonance eigemodes, which we will use for the energy exchange, have angular frequencies ω_1,2, frequency-widths due to intrinsic (absorption, radiation etc.) losses Γ_1,2 and (generally) vector fields F_1,2(r), normalized to unity energy. Once the two resonant objects are brought in proximity, they can interact and an appropriate analytical framework for modeling this resonant interaction is that of the well-known coupled-mode theory (CMT). In this picture, the field of the system of the two resonant objects 1, 2 can be approximated by F(r,t)=a₁(t)F₁(r)+a₂(t)F₂(r), where a_1,2(t) are the field amplitudes, with |a_1,2(t)|² equal to the energy stored inside the object 1, 2 respectively, due to the normalization. Then, using e^−iωt time dependence, the field amplitudes can be shown to satisfy, to lowest order:

[0078] $\begin{array}{cc}\frac{d}{dt}{a}_1\left(t\right)=-i\left({\omega }_1-i{\Gamma }_1\right)a_1\left(t\right)+i{\kappa }_{11}{a}_1\left(t\right)+i{\kappa }_{12}{a}_2\left(t\right)\frac{d}{dt}{a}_2\left(t\right)=-i\left({\omega }_2-i{\Gamma }_2\right)a_2\left(t\right)+i{\kappa }_{21}{a}_1\left(t\right)+i{\kappa }_{22}{a}_2\left(t\right)& \left(1\right)\end{array}$
where κ_11,22 are the shifts in each object's frequency due to the presence of the other, which are a second-order correction and can be absorbed into the eigenfrequencies by setting ω_1,2→ω_1,2+κ_11,22, and κ_12,21 are the coupling coefficients, which from the reciprocity requirement of the system must satisfy κ₂₁=κ₁₂≡κ.

[0079] The normal modes of the combined system are found, by substituting [a₁(t), a₂(t)]=[A₁, A₂]e^{−i ωt}, to have complex frequencies

[0080] $\begin{array}{cc}{\stackrel{\_}{\omega }}_{\pm }=\frac{{\omega }_1+{\omega }_2}{2}-i\frac{{\Gamma }_1+{\Gamma }_2}{2}\pm \sqrt{{\left(\frac{{\omega }_1-{\omega }_2}{2}-i\frac{{\Gamma }_1-{\Gamma }_2}{2}\right)}^2+{\kappa }^2}& \left(2\right)\end{array}$
whose splitting we denote as δ_E≡ ω₊− ω₋. Note that, at exact resonance ω₁=ω₂ and for Γ₁=Γ₂, we get δ_E=2κ.

[0081] Assume now that at time t=0 the source object 1 has finite energy |a₁(0)|², while the device object has |a₂(0)|²=0. Since the objects are coupled, energy will be transferred from 1 to 2. With these initial conditions, Eqs. (1) can be solved, predicting the evolution of the device field-amplitude to be

[0082] $\begin{array}{cc}\frac{a_2\left(t\right)}{a_1\left(0\right)}=\frac{2\kappa }{{\delta }_E}\sin \left(\frac{{\delta }_{E}t}{2}\right)e^{-\frac{{\Gamma }_1+{\Gamma }_2}{2}t}.& \left(3\right)\end{array}$
The energy-transfer efficiency will be η_E≡|a₂(t)|²/|a₁(0)|². Note that, at exact resonance ω₁=ω₂ and in the special case Γ₁=Γ₂≡Γ_o, Eq. (3) can be written as

[0083] $\begin{array}{cc}\frac{a_2\left(T\right)}{a_1\left(0\right)}=\sin \left(\mathrm{UT}\right)·e^{-T}& \left(4\right)\end{array}$
where T≡Γ_ot and U=κ/Γ_o.

[0084] In some examples, the system designer can adjust the duration of the coupling t at will. In some examples, the duration t can be adjusted to maximize the device energy (and thus efficiency η_E). Then, in the special case Γ₁=Γ₂=Γ_o, it can be inferred from Eq. (4) that η_E is maximized for

[0085] $\begin{array}{cc}{T}_{*}=\frac{{\tan }^{-1}U}{U}& \left(5\right)\end{array}$
resulting in an optimal energy-transfer efficiency

[0086] $\begin{array}{cc}{\eta }_{E^{*}}\equiv {\eta }_E\left(T_{*}\right)=\frac{U^2}{1+U^2}\exp \left(-\frac{2{\tan }^{-1}U}{U}\right)& \left(6\right)\end{array}$
which is only a function of the coupling-to-loss ratio U=κ/Γ_o and tends to unity when U>>1, as depicted in FIG. 2(c). In general, also for Γ₁≠Γ₂, the energy transfer is nearly perfect, when the coupling rate is much faster than all loss rates (κ/Γ_1,2>>1).

[0087] In a real wireless energy-transfer system, the source object can be connected to a power generator (not shown in FIG. 1), and the device object can be connected to a power consuming load (e.g. a resistor, a battery, an actual device, not shown in FIG. 1). The generator will supply the energy to the source object, the energy will be transferred wirelessly and non-radiatively from the source object to the device object, and the load will consume the energy from the device object. To incorporate such supply and consumption mechanisms into this temporal scheme, in some examples, one can imagine that the generator is very briefly but very strongly coupled to the source at time t=0 to almost instantaneously provide the energy, and the load is similarly very briefly but very strongly coupled to the device at the optimal time t=t* to almost instantaneously drain the energy. For a constant powering mechanism, at time t=t* also the generator can again be coupled to the source to feed a new amount of energy, and this process can be repeated periodically with a period t*.

1.2 Finite-Rate Energy-Transfer (Power-Transmission) Efficiency

[0088] Let the generator be continuously supplying energy to the source object 1 at a rate κ₁ and the load continuously draining energy from the device object 2 at a rate κ₂. Field amplitudes s_±1,2(t) are then defined, so that |s_±1,2(t)|² is equal to the power ingoing to (for the + sign) or outgoing from (for the − sign) the object 1, 2 respectively, and the CMT equations are modified to

[0089] $\begin{array}{cc}\frac{d}{dt}{a}_1\left(t\right)=-i\left({\omega }_1-i{\Gamma }_1\right)a_1\left(t\right)+i{\kappa }_{11}{a}_1\left(t\right)+i{\kappa }_{12}{a}_2\left(t\right)-{\kappa }_1{a}_1\left(t\right)+\sqrt{2{\kappa }_1}{s}_{+1}\left(t\right)\frac{d}{dt}{a}_2\left(t\right)=-i\left({\omega }_2-i{\Gamma }_2\right)a_2\left(t\right)+i{\kappa }_{21}{a}_1\left(t\right)+i{\kappa }_{22}{a}_2\left(t\right)-{\kappa }_2{a}_2\left(t\right)s_{-1}\left(t\right)=\sqrt{2{\kappa }_1}{a}_1\left(t\right)-s_{+1}\left(t\right)s_{-2}\left(t\right)=\sqrt{2{\kappa }_2}{a}_2\left(t\right)& \left(7\right)\end{array}$
where again we can set ω_1,2→ω_1,2+κ_11,22 and κ₂₁=κ₁₂≡K.

[0090] Assume now that the excitation is at a fixed frequency ω, namely has the form s₊₁(t)=S₊₁e^1ωt. Then the response of the linear system will be at the same frequency, namely a_1,2(t)=A_1,2e^−iωt and s_−1,2(t)=S_−1,2e^−iωt. By substituting these into Eqs. (7), using δ_1,2≡ω−ω_1,2, and solving the system, we find the field-amplitude transmitted to the load (S₂₁ scattering-matrix element)

[0091] $\hspace{1em}\begin{array}{cc}\begin{array}{c}{S}_{21}\equiv \frac{S_{-2}}{S_{+1}^{*}}\\ =\frac{2i\kappa \sqrt{{\kappa }_1{\kappa }_2}}{\left({\Gamma }_1+{\kappa }_1-i{\delta }_1\right)\left({\Gamma }_2+{\kappa }_2-i{\delta }_2\right)+{\kappa }^2}\\ =\frac{2iU\sqrt{U_1{U}_2}}{\left(1+U_1-i{D}_1\right)\left(1+U_2-i{D}_2\right)+U^2}.\end{array}& \left(8\right)\end{array}$
and the field-amplitude reflected to the generator (S₁₁ scattering-matrix element)

[0092] $\hspace{1em}\begin{array}{cc}\begin{array}{c}{S}_{11}\equiv \frac{S_{-1}}{S_{+1}}\\ =\frac{\left({\Gamma }_1-{\kappa }_1-i{\delta }_1\right)\left({\Gamma }_2+{\kappa }_2-i{\delta }_2\right)+{\kappa }^2}{\left({\Gamma }_1+{\kappa }_1-i{\delta }_1\right)\left({\Gamma }_2+{\kappa }_2-i{\delta }_2\right)+{\kappa }^2}\\ =\frac{\left(1-U_1-i{D}_1\right)\left(1+U_2-i{D}_2\right)+U^2}{\left(1+U_1-i{D}_1\right)\left(1+U_2-i{D}_2\right)+U^2}\end{array}& \left(9\right)\end{array}$
where D_1,2≡δ_1,2/Γ_1,2, U_1,2≡κ_1,2/Γ_1,2 and U≡κ/√{square root over (Γ₁Γ₂)}. Similarly, the scattering-matrix elements S₁₂, S₂₂ are given by interchanging 12 in Eqs. (8), (9) and, as expected from reciprocity, S₂₁=S₁₂. The coefficients for power transmission (efficiency) and reflection and loss are respectively η_P≡|S₂₁|²=|S₋₂|²/|S_+1|² and |S₁₁|²=|S₋₁|²/|S₊₁|² and 1−|S_21|²−|S₁₁|²=(2 Γ₁|A₁|²+2Γ₂|A₂|²)|S₊₁|².

[0093] In practice, in some implementations, the parameters D_1,2, U_1,2 can be designed (engineered), since one can adjust the resonant frequencies ω_1,2 (compared to the desired operating frequency ω) and the generator/load supply/drain rates κ_1,2 Their choice can target the optimization of some system performance-characteristic of interest:

[0094] In some examples, a goal can be to maximize the power transmission (efficiency) η_P≡|S₂₁|² of the system, so one would require
η_P(D_1,2)=η_P(U_1,2)=0   (10)

[0095] Since S₂₁ (from Eq. (8)) is symmetric upon interchanging 12, the optimal values for D_1,2 (determined by Eqs. (10)) will be equal, namely D₁=D₂≡D_o, and similarly U₁=U₂≡U_o. Then,

[0096] $\begin{array}{cc}{S}_{21}=\frac{2i{\mathrm{UU}}_o}{{\left(1+U_o-i{D}_o\right)}^2+U^2}& \left(11\right)\end{array}$
and from the condition η′_P(D_o)=0 we get that, for fixed values of U and U_o, the efficiency can be maximized for the following values of the symmetric detuning

[0097] $\begin{array}{cc}{D}_o=\begin{array}{cc}\pm \sqrt{U^2-{\left(1+U_o\right)}^2},& \mathrm{if}U>1+U_o\\ 0,& \mathrm{if}U\le 1+U_o\end{array},& \left(12\right)\end{array}$
which, in the case U>1+U_o, can be rewritten for the two frequencies at which the efficiency peaks as

[0098] $\begin{array}{cc}{\omega }_{\pm }=\frac{{\omega }_1{\Gamma }_2+{\omega }_2{\Gamma }_1}{{\Gamma }_1+{\Gamma }_2}\pm \frac{2\sqrt{{\Gamma }_1{\Gamma }_2}}{{\Gamma }_1+{\Gamma }_2}\sqrt{{\kappa }^2-\left({\Gamma }_1+{\kappa }_1\right)\left({\Gamma }_2+{\kappa }_2\right)},& \left(13\right)\end{array}$
whose splitting we denote as δ_P≡ ω₊− ω₋. Note that, at exact resonance ω₁=ω₂, and for Γ₁=Γ₂≡Γ_o and κ₁=κ₂≡κ_o, we get δ_P=2√{square root over (κ²−(Γ_o+κ_o)²)}<δ_E, namely the transmission-peak splitting is smaller than the normal-mode splitting. Then, by substituting D_o into η_P from Eq. (12), from the condition η′_P(U_o)=0 we get that, for fixed value of U, the efficiency can be maximized for
U_o*=√{square root over (1+²)} D_o*=0   (14)
which is known as ‘critical coupling’ condition, whereas for U_o<U_o*, the system is called ‘undercoupled’ and for U_o>U_o*. it is called ‘overcoupled’. The dependence of the efficiency on the frequency detuning D_o for different values of U_o (including the ‘critical-coupling’ condition) are shown in FIG. 2(a,b). The overall optimal power efficiency using Eqs. (14) is

[0099] $\begin{array}{cc}{\eta }_{P^{*}}\equiv {\eta }_P\left(D_{o^{*}},U_{o^{*}}\right)=\frac{U_{o^{*}}-1}{U_{o^{*}}+1}={\left(\frac{U}{1+\sqrt{1+U^2}}\right)}^2,& \left(15\right)\end{array}$
which is again only a function of the coupling-to-loss ratio U=κ/√{square root over (Γ₁Γ₂)} and tends to unity when U>>1, as depicted in FIG. 2(c).

[0100] In some examples, a goal can be to minimize the power reflection at the side of the generator |S₁₁|² and the load |S₂₂|², so one would then need
S_11,22=0(1∓U₁−iD₁)(1±U₂−iD₂)+U²=0,   (16)

[0101] The equations above present ‘impedance matching’ conditions. Again, the set of these conditions is symmetric upon interchanging 12, so, by substituting D₁=D₂≡D_o and U₁=U₂≡U_o into Eqs. (16), we get
(1−iD_o)²−U_o²+U²=0,   (17)
from which we easily find that the values of D_o and U_o that cancel all reflections are again exactly those in Eqs. (14).

[0102] It can be seen that, for this particular problem, the two goals and their associated sets of conditions (Eqs. (10) and Eqs. (16)) result in the same optimized values of the intra-source and intra-device parameters D_1,2, U_1,2. Note that for a lossless system this would be an immediate consequence of power conservation (Hermiticity of the scattering matrix), but this is not apparent for a lossy system.

[0103] Accordingly, for any temporal energy-transfer scheme, once the parameters specific only to the source or to the device (such as their resonant frequencies and their excitation or loading rates respectively) have been optimally designed, the efficiency monotonically increases with the ratio of the source-device coupling-rate to their loss rates. Using the definition of a resonance quality factor Q=ω/2Γ and defining by analogy the coupling factor k≡1/Q_κ≡2κ/√{square root over (ω₁ω₂)}, it is therefore exactly this ratio

[0104] $\begin{array}{cc}U=\frac{\kappa }{\sqrt{{\Gamma }_1{\Gamma }_2}}=k\sqrt{Q_1{Q}_2}& \left(18\right)\end{array}$
that has been set as a figure-of-merit for any system under consideration for wireless energy-transfer, along with the distance over which this ratio can be achieved (clearly, U will be a decreasing function of distance). The desired optimal regime U>1 is called ‘strong-coupling’ regime and it is a necessary and sufficient condition for efficient energy-transfer. In particular, for U>1 we get, from Eq. (15), η_P*>17%, large enough for practical applications. The figure-of-merit U is called the strong-coupling factor. We will further show how to design systems with a large strong-coupling factor.

[0105] To achieve a large strong-coupling factor U, in some examples, the energy-transfer application preferably uses resonant modes of high quality factors Q, corresponding to low (i.e. slow) intrinsic-loss rates Γ. This condition can be satisfied by designing resonant modes where all loss mechanisms, typically radiation and absorption, are sufficiently suppressed.

[0106] This suggests that the coupling be implemented using, not the lossy radiative far-field, which should rather be suppressed, but the evanescent (non-lossy) stationary near-field. To implement an energy-transfer scheme, usually more appropriate are finite objects, namely ones that are topologically surrounded everywhere by air, into where the near field extends to achieve the coupling. Objects of finite extent do not generally support electromagnetic states that are exponentially decaying in all directions in air away from the objects, since Maxwell's Equations in free space imply that k²=ω²/c₂, where k is the wave vector, ω the angular frequency, and c the speed of light, because of which one can show that such finite objects cannot support states of infinite Q, rather there always is some amount of radiation. However, very long-lived (so-called “high-Q”) states can be found, whose tails display the needed exponential or exponential-like decay away from the resonant object over long enough distances before they turn oscillatory (radiative). The limiting surface, where this change in the field behavior happens, is called the “radiation caustic”, and, for the wireless energy-transfer scheme to be based on the near field rather than the far/radiation field, the distance between the coupled objects must be such that one lies within the radiation caustic of the other. One typical way of achieving a high radiation-Q (Q_rad) is to design subwavelength resonant objects. When the size of an object is much smaller than the wavelength of radiation in free space, its electromagnetic field couples to radiation very weakly. Since the extent of the near-field into the area surrounding a finite-sized resonant object is set typically by the wavelength, in some examples, resonant objects of subwavelength size have significantly longer evanescent field-tails. In other words, the radiation caustic is pushed far away from the object, so the electromagnetic mode enters the radiative regime only with a small amplitude.

[0107] Moreover, most realistic materials exhibit some nonzero amount of absorption, which can be frequency dependent, and thus cannot support states of infinite Q, rather there always is some amount of absorption. However, very long-lived (“high-Q”) states can be found, where electromagnetic modal energy is only weakly dissipated. Some typical ways of achieving a high absorption-Q (Q_abs) is to use materials which exhibit very small absorption at the resonant frequency and/or to shape the field to be localized more inside the least lossy materials.

[0108] Furthermore, to achieve a large strong-coupling factor U, in some examples, the energy-transfer application preferably uses systems that achieve a high coupling factor k, corresponding to strong (i.e. fast) coupling rate κ, over distances larger than the characteristic sizes of the objects.

[0109] Since finite-sized subwavelength resonant objects can often be accompanied with a high Q, as was discussed above and will be seen in examples later on, such an object will typically be the appropriate choice for the possibly-mobile resonant device-object. In these cases, the electromagnetic field is, in some examples, of quasi-static nature and the distance, up to which sufficient coupling can be achieved, is dictated by the decay-law of this quasi-static field.

[0110] Note, though, that in some examples, the resonant source-object will be immobile and thus less restricted in its allowed geometry and size. It can be therefore chosen large enough that the near-field extent is not limited by the wavelength, and can thus have nearly infinite radiation-Q. Some objects of nearly infinite extent, such as dielectric waveguides, can support guided modes, whose evanescent tails are decaying exponentially in the direction away from the object, slowly if tuned close to cutoff, therefore a good coupling can also be achieved over distances quite a few times larger than a characteristic size of the source- and/or device-object.

2 ‘Strongly-Coupled’ Resonances at Mid-Range Distances for Realistic Systems

[0111] In the following, examples of systems suitable for energy transfer of the type described above are described. We will demonstrate how to compute the CMT parameters ω_1,2, Q_1,2 and k described above and how to choose or design these parameters for particular examples in order to produce a desirable figure-of-merit U=κ/√{square root over (Γ₁Γ₂)}=k√{square root over (Q₁Q₂)} at a desired distance D. In some examples, this figure-of-merit is maximized when ω_1,2 are tuned close to a particular angular frequency ω_U.

2.1 Self-Resonant Conducting Coils

[0112] In some examples, one or more of the resonant objects are self-resonant conducting coils. Referring to FIG. 3, a conducting wire of length 1 and cross-sectional radius a is wound into a helical coil of radius r and height h (namely with N=√{square root over (l²−h²)}/2πr number of turns), surrounded by air. As described below, the wire has distributed inductance and distributed capacitance, and therefore it supports a resonant mode of angular frequency ω. The nature of the resonance lies in the periodic exchange of energy from the electric field within the capacitance of the coil, due to the charge distribution ρ(x) across it, to the magnetic field in free space, due to the current distribution j(x) in the wire. In particular, the charge conservation equation ∇·j=iωρ implies that: (i) this periodic exchange is accompanied by a π/2 phase-shift between the current and the charge density profiles, namely the energy W contained in the coil is at certain points in time completely due to the current and at other points in time completely due to the charge, and (ii) if ρ_l(x) and I(x) are respectively the linear charge and current densities in the wire, where x runs along the wire,

[0113] $q_o=\frac{1}{2}\int dx{\rho }_l\left(x\right)$
is the maximum amount of positive charge accumulated in one side of the coil (where an equal amount of negative charge always also accumulates in the other side to make the system neutral) and I_o=max{|I(x)|} is the maximum positive value of the linear current distribution, then I_o=ωq_o. Then, one can define an effective total inductance L and an effective total capacitance C of the coil through the amount of energy W inside its resonant mode:

[0114] $\begin{array}{cc}W\equiv \frac{1}{2}{I}_o^{2}L\Rightarrow L=\frac{{\mu }_o}{4\pi I_o^2}\int \int dxd{x}^{\prime }\frac{j\left(x\right)·j\left(x^{\prime }\right)}{x-x^{\prime }},& \left(19\right)\\ W\equiv \frac{1}{2}{q}_o^2\frac{1}{C}\Rightarrow \frac{1}{C}=\frac{1}{4\pi {\varepsilon }_o{q}_o^2}\int \int dxd{x}^{\prime }\frac{\rho \left(x\right)\rho \left(x^{\prime }\right)}{x-x^{\prime }},& \left(20\right)\end{array}$
where μ_o and ∈_o are the magnetic permeability and electric permittivity of free space.

[0115] With these definitions, the resonant angular frequency and the effective impedance can be given by the formulas ω=1/√{square root over (LC)} and Z=√{square root over (L/C)} respectively.

[0116] Losses in this resonant system consist of ohmic (material absorption) loss inside the wire and radiative loss into free space. One can again define a total absorption resistance R_abs from the amount of power absorbed inside the wire and a total radiation resistance R_rad from the amount of power radiated due to electric- and magnetic-dipole radiation:

[0117] $\begin{array}{cc}{P}_{\mathrm{abs}}\equiv \frac{1}{2}{I}_o^2{R}_{\mathrm{abs}}\Rightarrow R_{\mathrm{abs}}\approx {\zeta }_c\frac{l}{2\pi a}·\frac{I_{\mathrm{rms}}^2}{I_o^2}& \left(21\right)\\ P_{\mathrm{rad}}\equiv \frac{1}{2}{I}_o^2{R}_{\mathrm{rad}}\Rightarrow R_{\mathrm{rad}}\approx \frac{{\zeta }_o}{6\pi }\left[{\left(\frac{\omega p}{c}\right)}^2+{\left(\frac{\omega \sqrt{m}}{c}\right)}^4\right],& \left(22\right)\end{array}$
where c=1/√{square root over (μ_o∈_o)} and ζ_o=√{square root over (μ_o/∈_o)} are the light velocity and light impedance in free space, the impedance ζ_c is ζ_c=1/σδ=√{square root over (μ_oω/2σ)} with σ the conductivity of the conductor and δ the skin depth at the frequency ω,

[0118] $I_{\mathrm{rms}}^2=\frac{1}{l}\int dx{I\left(x\right)}^2,p=\int dxr{\rho }_l\left(x\right)$
is the electric-dipole moment of the coil and

[0119] $m=\frac{1}{2}\int dxr\times j\left(x\right)$
is the magnetic-dipole moment of the coil. For the radiation resistance formula Eq. (22), the assumption of operation in the quasi-static regime (h, r<<λ=2πc/ω) has been used, which is the desired regime of a subwavelength resonance. With these definitions, the absorption and radiation quality factors of the resonance are given by Q_abs=Z/R_abs and Q_rad=Z/R_rad respectively.

[0120] From Eq. (19)-(22) it follows that to determine the resonance parameters one simply needs to know the current distribution j in the resonant coil. Solving Maxwell's equations to rigorously find the current distribution of the resonant electromagnetic eigenmode of a conducting-wire coil is more involved than, for example, of a standard LC circuit, and we can find no exact solutions in the literature for coils of finite length, making an exact solution difficult. One could in principle write down an elaborate transmission-line-like model, and solve it by brute force. We instead present a model that is (as described below) in good agreement (˜5%) with experiment. Observing that the finite extent of the conductor forming each coil imposes the boundary condition that the current has to be zero at the ends of the coil, since no current can leave the wire, we assume that the resonant mode of each coil is well approximated by a sinusoidal current profile along the length of the conducting wire. We shall be interested in the lowest mode, so if we denote by x the coordinate along the conductor, such that it runs from −l/2 to +l/2, then the current amplitude profile would have the form I(x)=I_o cos(πx/l), where we have assumed that the current does not vary significantly along the wire circumference for a particular x, a valid assumption provided a<<r. It immediately follows from the continuity equation for charge that the linear charge density profile should be of the form ρ_l(x)=ρ_o sin (πx/l), and thus q_o32 ∫₀^l/2dxρ_o|sin(πx/l)|=ρ_ol/π. Using these sinusoidal profiles we find the so-called “self-inductance” L_s and “self-capacitance” C_s of the coil by computing numerically the integrals Eq. (19) and (20); the associated frequency and effective impedance are ω_s and Z_s respectively. The “self-resistances” R_s are given analytically by Eq. (21) and (22) using

[0121] $I_{\mathrm{rms}}^2=\frac{1}{l}{\int }_{-l/2}^{l/2}dx{I_o\cos \left(\pi x/l\right)}^2=\frac{1}{2}{I}_o^2,p=q_o\sqrt{{\left(\frac{2}{\pi }h\right)}^2+{\left(\frac{4N\cos \left(\pi N\right)}{\left(4N^2-1\right)\pi }r\right)}^2}$
and

[0122] $m=I_o\sqrt{{\left(\frac{2}{\pi }N\pi r^2\right)}^2+{\left(\frac{\cos \left(\pi N\right)\left(12N^2-1\right)-\sin \left(\pi N\right)\pi N\left(4N^2-1\right)}{\left(16N^4-8N^2+1\right)\pi }\mathrm{hr}\right)}^2},$
and therefore the associated Q_s factors can be calculated.

[0123] The results for two examples of resonant coils with subwavelength modes of λ_s/r≧70 (i.e. those highly suitable for near-field coupling and well within the quasi-static limit) are presented in Table 1. Numerical results are shown for the wavelength and absorption, radiation and total loss rates, for the two different cases of subwavelength-coil resonant modes. Note that, for conducting material, copper (σ=5.998·10^−7 S/m) was used. It can be seen that expected quality factors at microwave frequencies are Q_s,abs≧1000 and Q_s,rad≧5000.

[0124]

TABLE 1
single coil	λs/r	f (MHz)	Qs,rad	Qs,abs	Qs
r = 30 cm, h = 20 cm, a = 1 cm,	74.7	13.39	4164	8170	2758
N = 4
r = 10 cm, h = 3 cm, a = 2 mm,	140	21.38	43919	3968	3639
N = 6

[0125] Referring to FIG. 4, in some examples, energy is transferred between two self-resonant conducting-wire coils. The electric and magnetic fields are used to couple the different resonant conducting-wire coils at a distance D between their centers. Usually, the electric coupling highly dominates over the magnetic coupling in the system under consideration for coils with h>>2r and, oppositely, the magnetic coupling highly dominates over the electric coupling for coils with h<<2r. Defining the charge and current distributions of two coils 1,2 respectively as ρ_1,2(x) and j_1,2(x), total charges and peak currents respectively as q_1,2 and I_1,2, and capacitances and inductances respectively as C_1,2 and L_1,2 which are the analogs of ρ(x), j(x), q_o, I_o, C and L for the single-coil case and are therefore well defined, we can define their mutual capacitance and inductance through the total energy:

[0126] $\begin{array}{cc}W\equiv W_1+W_2+\frac{1}{2}\left(q_1^{*}{q}_2+q_2^{*}{q}_1\right)/M_C+\frac{1}{2}\left(I_1^{*}{I}_2+I_2^{*}{I}_1\right)M_L\Rightarrow 1/M_C=\frac{1}{4{\mathrm{\pi \varepsilon }}_o{q}_1{q}_2}\int \int dxd{x}^{\prime }\frac{{\rho }_1\left(x\right){\rho }_2\left(x^{\prime }\right)}{x-x^{\prime }}u,M_L=\frac{{\mu }_o}{4\pi I_1{I}_2}\int \int dxd{x}^{\prime }\frac{j_1\left(x\right)·j_2\left(x^{\prime }\right)}{x-x^{\prime }}u,& \left(23\right)\end{array}$
where

[0127] $W_1=\frac{1}{2}{q}_1^2/C_1=\frac{1}{2}{I}_1^2{L}_1,W_2=\frac{1}{2}{q}_2^2/C_2=\frac{1}{2}{I}_2^2{L}_2$
and the retardation factor of u=exp(iω|x−x′|/c) inside the integral can been ignored in the quasi-static regime D<<λ of interest, where each coil is within the near field of the other. With this definition, the coupling factor is given by k=√{square root over (C₁C₂)}/M_C+M_L/√{square root over (L₁L₂)}.

[0128] Therefore, to calculate the coupling rate between two self-resonant coils, again the current profiles are needed and, by using again the assumed sinusoidal current profiles, we compute numerically from Eq. (23) the mutual capacitance M_C,s and inductance M_L,s between two self-resonant coils at a distance D between their centers, and thus k=1/Q_κ is also determined.

[0129]

TABLE 2
	pair of coils	D/r	Q	Qκ = 1/k	U
	r = 30 cm, h = 20 cm,	3	2758	38.9	70.9
	a = 1 cm, N = 4	5	2758	139.4	19.8
	λ/r ≈ 75	7	2758	333.0	8.3
	Qsabs ≈ 8170, Qsrad ≈ 4164	10	2758	818.9	3.4
	r = 10 cm, h = 3 cm,	3	3639	61.4	59.3
	a = 2 mm, N = 6	5	3639	232.5	15.7
	λ/r ≈ 140	7	3639	587.5	6.2
	Qsabs ≈ 3968, Qsrad ≈ 43919	10	3639	1580	2.3

[0130] Referring to Table 2, relevant parameters are shown for exemplary examples featuring pairs or identical self resonant coils. Numerical results are presented for the average wavelength and loss rates of the two normal modes (individual values not shown), and also the coupling rate and figure-of-merit as a function of the coupling distance D, for the two cases of modes presented in Table 1. It can be seen that for medium distances D/r=10−3 the expected coupling-to-loss ratios are in the range U˜2-70.

2.1.1 Experimental Results

[0131] An experimental realization of an example of the above described system for wireless energy transfer consists of two self-resonant coils of the type described above, one of which (the source coil) is coupled inductively to an oscillating circuit, and the second (the device coil) is coupled inductively to a resistive load, as shown schematically in FIG. 5. Referring to FIG. 5, A is a single copper loop of radius 25 cm that is part of the driving circuit, which outputs a sine wave with frequency 9.9 MHz. s and d are respectively the source and device-coils referred to in the text. B is a loop of wire attached to the load (“light-bulb”). The various κ's represent direct couplings between the objects. The angle between coil d and the loop A is adjusted so that their direct coupling is zero, while coils s and d are aligned coaxially. The direct coupling between B and A and between B and s is negligible.

[0132] The parameters for the two identical helical coils built for the experimental validation of the power transfer scheme were h=20 cm, a=3 mm, r=30 cm and N=5.25. Both coils are made of copper. Due to imperfections in the construction, the spacing between loops of the helix is not uniform, and we have encapsulated the uncertainty about their uniformity by attributing a 10% (2 cm) uncertainty to h. The expected resonant frequency given these dimensions is f₀=10.56±0.3 MHz, which is about 5% off from the measured resonance at around 9.90 MHz.

[0133] The theoretical Q for the loops is estimated to be ˜2500 (assuming perfect copper of resistivity ρ=1/σ=1.7×10⁻⁸ Ωm) but the measured value is 950±50. We believe the discrepancy is mostly due to the effect of the layer of poorly conducting copper oxide on the surface of the copper wire, to which the current is confined by the short skin depth (˜20 μm) at this frequency. We have therefore used the experimentally observed Q (and Γ₁=Γ₂=Γ=ω/(2Q) derived from it) in all subsequent computations.

[0134] The coupling coefficient κ can be found experimentally by placing the two self-resonant coils (fine-tuned, by slightly adjusting h, to the same resonant frequency when isolated) a distance D apart and measuring the splitting in the frequencies of the two resonant modes in the transmission spectrum. According to Eq. (13) derived by coupled-mode theory, the splitting in the transmission spectrum should be δ_P=2√{square root over (κ²−Γ²)}, when κ_A,B are kept very small by keeping A and B at a relatively large distance. The comparison between experimental and theoretical results as a function of distance when the two the coils are aligned coaxially is shown in FIG. 6.

[0135] FIG. 7 shows a comparison of experimental and theoretical values for the strong-coupling factor U=κ/Γ as a function of the separation between the two coils. The theory values are obtained by using the theoretically obtained κ and the experimentally measured Γ. The shaded area represents the spread in the theoretical U due to the ˜5% uncertainty in Q. As noted above, the maximum theoretical efficiency depends only on the parameter U, which is plotted as a function of distance in FIG. 7. U is greater than 1 even for D=2.4 m (eight times the radius of the coils), thus the system is in the strongly-coupled regime throughout the entire range of distances probed.

[0136] The power-generator circuit was a standard Colpitts oscillator coupled inductively to the source coil by means of a single loop of copper wire 25 cm in radius (see FIG. 5). The load consisted of a previously calibrated light-bulb, and was attached to its own loop of insulated wire, which was in turn placed in proximity of the device coil and inductively coupled to it. Thus, by varying the distance between the light-bulb and the device coil, the parameter U_B=κ_B/Γ was adjusted so that it matched its optimal value, given theoretically by Eq. (14) as U_B*=√{square root over (1+U²)}. Because of its inductive nature, the loop connected to the light-bulb added a small reactive component to κ_B which was compensated for by slightly retuning the coil. The work extracted was determined by adjusting the power going into the Colpitts oscillator until the light-bulb at the load was at its full nominal brightness.

[0137] In order to isolate the efficiency of the transfer taking place specifically between the source coil and the load, we measured the current at the mid-point of each of the self-resonant coils with a current-probe (which was not found to lower the Q of the coils noticeably.) This gave a measurement of the current parameters I₁ and I₂ defined above. The power dissipated in each coil was then computed from P_1,2=ΓL|I_1,2|², and the efficiency was directly obtained from η=P_B/(P₁+P₂+P_B). To ensure that the experimental setup was well described by a two-object coupled-mode theory model, we positioned the device coil such that its direct coupling to the copper loop attached to the Colpitts oscillator was zero. The experimental results are shown in FIG. 8, along with the theoretical prediction for maximum efficiency, given by Eq. (15).

[0138] Using this example, we were able to transmit significant amounts of power using this setup from the source coil to the device coil, fully lighting up a 60 W light-bulb from distances more than 2 m away, for example. As an additional test, we also measured the total power going into the driving circuit. The efficiency of the wireless power-transmission itself was hard to estimate in this way, however, as the efficiency of the Colpitts oscillator itself is not precisely known, although it is expected to be far from 100%. Nevertheless, this gave an overly conservative lower bound on the efficiency. When transmitting 60 W to the load over a distance of 2 m, for example, the power flowing into the driving circuit was 400 W. This yields an overall wall-to-load efficiency of ˜15%, which is reasonable given the expected ˜40% efficiency for the wireless power transmission at that distance and the low efficiency of the driving circuit.

[0139] From the theoretical treatment above, we see that in typical examples it is important that the coils be on resonance for the power transmission to be practical. We found experimentally that the power transmitted to the load dropped sharply as one of the coils was detuned from resonance. For a fractional detuning Δf/f₀ of a few times the inverse loaded Q, the induced current in the device coil was indistinguishable from noise.

[0140] The power transmission was not found to be visibly affected as humans and various everyday objects, such as metallic and wooden furniture, as well as electronic devices large and small, were placed between the two coils, even when they drastically obstructed the line of sight between source and device. External objects were found to have an effect only when they were closer than 10 cm from either one of the coils. While some materials (such as aluminum foil, styrofoam and humans) mostly just shifted the resonant frequency, which could in principle be easily corrected with a feedback circuit of the type described earlier, others (cardboard, wood, and PVC) lowered Q when placed closer than a few centimeters from the coil, thereby lowering the efficiency of the transfer.

[0141] This method of power transmission is believed safe for humans. When transmitting 60 W (more than enough to power a laptop computer) across 2 m, we estimated that the magnitude of the magnetic field generated is much weaker than the Earth's magnetic field for all distances except for less than about 1 cm away from the wires in the coil, an indication of the safety of the scheme even after long-term use. The power radiated for these parameters was ˜5 W, which is roughly an order of magnitude higher than cell phones but could be drastically reduced, as discussed below.

[0142] Although the two coils are currently of identical dimensions, it is possible to make the device coil small enough to fit into portable devices without decreasing the efficiency. One could, for instance, maintain the product of the characteristic sizes of the source and device coils constant.

[0143] These experiments demonstrated experimentally a system for power transmission over medium range distances, and found that the experimental results match theory well in multiple independent and mutually consistent tests.

[0144] The efficiency of the scheme and the distances covered can be appreciably improved by silver-plating the coils, which should increase their Q, or by working with more elaborate geometries for the resonant objects. Nevertheless, the performance characteristics of the system presented here are already at levels where they could be useful in practical applications.

2.2 Capacitively-Loaded Conducting Loops or Coils

[0145] In some examples, one or more of the resonant objects are capacitively-loaded conducting loops or coils. Referring to FIG. 9 a helical coil with N turns of conducting wire, as described above, is connected to a pair of conducting parallel plates of area A spaced by distance d via a dielectric material of relative permittivity ∈, and everything is surrounded by air (as shown, N=1 and h=0). The plates have a capacitance C_p=∈_o∈A/d, which is added to the distributed capacitance of the coil and thus modifies its resonance. Note however, that the presence of the loading capacitor modifies significantly the current distribution inside the wire and therefore the total effective inductance L and total effective capacitance C of the coil are different respectively from L_s and C_s, which are calculated for a self-resonant coil of the same geometry using a sinusoidal current profile. Since some charge is accumulated at the plates of the external loading capacitor, the charge distribution ρ inside the wire is reduced, so C<C_s, and thus, from the charge conservation equation, the current distribution j flattens out, so L>L_s. The resonant frequency for this system is ω=1/√{square root over (L(C+C_p))}<ω_s=1/√{square root over (L_sC_s)}, and I(x)→I_o cos(πx/l)C→C_sω→ω_s, as C_p→0.

[0146] In general, the desired CMT parameters can be found for this system, but again a very complicated solution of Maxwell's Equations is required. Instead, we will analyze only a special case, where a reasonable guess for the current distribution can be made. When C_p>>C_s>C, then ω≈1/√{square root over (LC_p)}<<ω_s and Z≈√{square root over (L/C_p)}<<Z_s, while all the charge is on the plates of the loading capacitor and thus the current distribution is constant along the wire. This allows us now to compute numerically L from Eq. (19). In the case h=0 and N integer, the integral in Eq. (19) can actually be computed analytically, giving the formula L=μ_or[ln(8r/a)−2]N². Explicit analytical formulas are again available for R from Eq. (21) and (22), since I_rms=I_o, |p|≈0 and |m|=I_oNπr² (namely only the magnetic-dipole term is contributing to radiation), so we can determine also Q_abs=ωL/R_abs and Q_rad=ωL/R_rad. At the end of the calculations, the validity of the assumption of constant current profile is confirmed by checking that indeed the condition C_p>>C_sω<<ω_s is satisfied. To satisfy this condition, one could use a large external capacitance, however, this would usually shift the operational frequency lower than the optimal frequency, which we will determine shortly; instead, in typical examples, one often prefers coils with very small self-capacitance C_s to begin with, which usually holds, for the types of coils under consideration, when N=1, so that the self-capacitance comes from the charge distribution across the single turn, which is almost always very small, or when N>1 and h>>2Na, so that the dominant self-capacitance comes from the charge distribution across adjacent turns, which is small if the separation between adjacent turns is large.

[0147] The external loading capacitance C_p provides the freedom to tune the resonant frequency (for example by tuning A or d). Then, for the particular simple case h=0, for which we have analytical formulas, the total Q=ωL/(R_abs+R_rad) becomes highest at the optimal frequency

[0148] $\begin{array}{cc}{\omega }_Q={\left[\frac{c^4}{\pi }\sqrt{\frac{{\varepsilon }_o}{2\sigma }}·\frac{1}{{\mathrm{aNr}}^3}\right]}^{2/7},& \left(24\right)\end{array}$
reaching the value

[0149] $\begin{array}{cc}{Q}_{\max }=\frac{6}{7\pi }{\left(2{\pi }^2{\eta }_o\frac{\sigma a^2{N}^2}{r}\right)}^{3/7}·\left[\ln \left(\frac{8r}{a}\right)-2\right].& \left(25\right)\end{array}$

[0150] At lower frequencies it is dominated by ohmic loss and at higher frequencies by radiation. Note, however, that the formulas above are accurate as long as ω_Q<<ω_s and, as explained above, this holds almost always when N=1, and is usually less accurate when N>1, since h=0 usually implies a large self-capacitance. A coil with large h can be used, if the self-capacitance needs to be reduced compared to the external capacitance, but then the formulas for L and ω_Q, Q_max are again less accurate. Similar qualitative behavior is expected, but a more complicated theoretical model is needed for making quantitative predictions in that case.

[0151] The results of the above analysis for two examples of subwavelength modes of λ/r≧70 (namely highly suitable for near-field coupling and well within the quasi-static limit) of coils with N=1 and h=0 at the optimal frequency Eq. (24) are presented in Table 3. To confirm the validity of constant-current assumption and the resulting analytical formulas, mode-solving calculations were also performed using another completely independent method: computational 3D finite-element frequency-domain (FEFD) simulations (which solve Maxwell's Equations in frequency domain exactly apart for spatial discretization) were conducted, in which the boundaries of the conductor were modeled using a complex impedance ζ_c=√{square root over (μ_oω/2σ)} boundary condition, valid as long as ζ_c/ζ_o<<1 (<10⁻⁵ for copper in the microwave). Table 3 shows Numerical FEFD (and in parentheses analytical) results for the wavelength and absorption, radiation and total loss rates, for two different cases of subwavelength-loop resonant modes. Note that for conducting material copper (σ=5.998·10⁷ S/m) was used. Specific parameters of the plot in FIG. 4 are highlighted in bold in the table. The two methods (analytical and computational) are in good agreement and show that, in some examples, the optimal frequency is in the low-MHz microwave range and the expected quality factors are Q_abs≧1000 and Q_rad≧10000.

[0152]

TABLE 3
single coil	λ/r	f	Qrad	Qabs	Q
r = 30 cm, a = 2 cm	111.4 (112.4)	8.976 (8.897)	29546 (30512)	4886 (5117)	4193 (4381)
ε = 10, A = 138 cm2,
d = 4 mm
r = 10 cm, a = 2 mm	69.7 (70.4)	43.04 (42.61)	10702 (10727)	1545 (1604)	1350 (1395)
ε = 10, A = 3.14 cm2,
d = 1 mm

[0153] Referring to FIG. 10, in some examples, energy is transferred between two capacitively-loaded coils. For the rate of energy transfer between two capacitively-loaded coils 1 and 2 at distance D between their centers, the mutual inductance M_L can be evaluated numerically from Eq. (23) by using constant current distributions in the case ω<<ω_s. In the case h=0, the coupling is only magnetic and again we have an analytical formula, which, in the quasi-static limit r<<D<<λ and for the relative orientation shown in FIG. 10, is M_L≈πμ_o/2·(r₁r₂)²N₁N₂/D³, which means that k ∝(√{square root over (r₁r₂)}/D)³ is independent of the frequency ω and the number of turns N₁, N₂, Consequently, the resultant coupling figure-of-merit of interest is

[0154] $\begin{array}{cc}U=k\sqrt{Q_1{Q}_2}\approx {\left(\frac{\sqrt{r_1{r}_2}}{iD}\right)}^3·\frac{{\pi }^2{\eta }_o\frac{\sqrt{r_1{r}_2}}{\lambda }·N_1{N}_2}{\prod _{j=1,2}{\left(\sqrt{\frac{\pi {\eta }_o}{\lambda \sigma }}·\frac{r_j}{a_j}{N}_j+\frac{8}{3}{\pi }^5{{\eta }_o\left(\frac{r_j}{\lambda }\right)}^4{N}_j^2\right)}^{1/2}},& \left(26\right)\end{array}$
which again is more accurate for N₁=N₂=1.

[0155] From Eq. (26) it can be seen that the optimal frequency ω_U, where the figure-of-merit is maximized to the value U_max, is close to the frequency ω_Q1Q2 at which Q₁Q₂ is maximized, since k does not depend much on frequency (at least for the distances D<<λ of interest for which the quasi-static approximation is still valid). Therefore, the optimal frequency ω_U≈ω_Q1Q2 is mostly independent of the distance D between the two coils and lies between the two frequencies ω_Q1 and ω_Q2 at which the single-coil Q₁ and Q₂ respectively peak. For same coils, this optimal frequency is given by Eq. (24) and then the strong-coupling factor from Eq. (26) becomes

[0156] $\begin{array}{cc}{U}_{\max }={\mathrm{kQ}}_{\max }\approx {\left(\frac{r}{D}\right)}^3·\frac{3}{7}{\left(2{\pi }^2{\eta }_o\frac{\sigma a^2{N}^2}{r}\right)}^{3/7}.& \left(27\right)\end{array}$

[0157] In some examples, one can tune the capacitively-loaded conducting loops or coils, so that their angular eigenfrequencies are close to ω_U within Γ_U, which is half the angular frequency width for which U>U_max/2.

[0158] Referring to Table 4, numerical FEFD and, in parentheses, analytical results based on the above are shown for two systems each composed of a matched pair of the loaded coils described in Table 3. The average wavelength and loss rates are shown along with the coupling rate and coupling to loss ratio figure-of-merit U=κ/Γ as a function of the coupling distance D, for the two cases. Note that the average numerical Γ_rad shown are slightly different from the single-loop value of FIG. 3, analytical results for Γ_rad are not shown but the single-loop value is used. (The specific parameters corresponding to the plot in FIG. 10 are highlighted with bold in the table.) Again we chose N=1 to make the constant-current assumption a good one and computed M_L numerically from Eq. (23). Indeed the accuracy can be confirmed by their agreement with the computational FEFD mode-solver simulations, which give κ through the frequency splitting of the two normal modes of the combined system (δ_E=2κ from Eq. (4)). The results show that for medium distances D/r=10−3 the expected coupling-to-loss ratios are in the range U˜0.5-50.

[0159]

TABLE 4
pair of coils	D/r	Qrad	Q = ω/2Γ	Qκ = ω/2κ	κ/Γ
r = 30 cm, a = 2 cm	3	30729	4216	62.6 (63.7)	67.4 (68.7)
ε = 10, A = 138 cm2, d = 4 mm	5	29577	4194	235 (248)	17.8 (17.6)
λ/r ≈ 112	7	29128	4185	589 (646)	7.1 (6.8)
Qabs ≈ 4886	10	28833	4177	1539 (1828)	2.7 (2.4)
r = 10 cm, a = 2 mm	3	10955	1355	85.4 (91.3)	15.9 (15.3)
ε = 10, A = 3.14 cm2, d = 1 mm	5	10740	1351	313 (356)	4.32 (3.92)
λ/r ≈ 70	7	10759	1351	754 (925)	1.79 (1.51)
Qabs ≈ 1546	10	10756	1351	1895 (2617)	0.71 (0.53)

2:2.1 Derivation of Optimal Power-Transmission Efficiency

[0160] Referring to FIG. 11, to rederive and express Eq. (15) in terms of the parameters which are more directly accessible from particular resonant objects, such as the capacitively-loaded conducting loops, one can consider the following circuit-model of the system, where the inductances L_s, L_d represent the source and device loops respectively, R_s, R_d their respective losses, and C_s, C_d are the required corresponding capacitances to achieve for both resonance at frequency ω. A voltage generator V_g is considered to be connected to the source and a load resistance R_l to the device. The mutual inductance is denoted by M.

[0161] Then from the source circuit at resonance (ωL_s=1/ωC_s):

[0162] $\begin{array}{cc}{V}_g=I_s{R}_s-\mathrm{j\omega }{\mathrm{MI}}_d\Rightarrow \frac{1}{2}{V}_g^{*}{I}_s=\frac{1}{2}{I_s}^2{R}_s+\frac{1}{2}\mathrm{j\omega }{\mathrm{MI}}_d^{*}{I}_s,& \left(28\right)\end{array}$
and from the device circuit at resonance (ωL_d=1/ωC_d):
0=I_d(R_d+R_l)−jωMI_sjωMI_s=I_d(R_d+R_l)   (29)

[0163] So by substituting Eq. (29) to Eq. (28) and taking the real part (for time-averaged power) we get:

[0164] $\begin{array}{cc}{P}_g=\mathrm{Re}\left\{\frac{1}{2}{V}_g^{*}{I}_s\right\}=\frac{1}{2}{I_s}^2{R}_s+\frac{1}{2}{I_d}^2\left(R_d+R_l\right)=P_s+P_d+P_l,& \left(30\right)\end{array}$
where we identified the power delivered by the generator P_g=Re{V_g*I_s/2}, the power lost inside the source P_s=|I_s|²R_s/2, the power lost inside the device P_d=|I_d|²R_d/2 and the power delivered to the load P_l=|I_d|²R_l/2. Then, the power transmission efficiency is:

[0165] $\begin{array}{cc}\begin{array}{c}{\eta }_P\equiv \frac{P_l}{P_g}\\ =\frac{R_l}{{\frac{I_s}{I_d}}^2{R}_s+\left(R_d+R_l\right)}\\ \stackrel{\left(29\right)}{=}\frac{R_l}{\frac{{\left(R_d+R_l\right)}^2}{{\left(\omega M\right)}^2}{R}_s+\left(R_d+R_l\right)}.\end{array}& \left(31\right)\end{array}$

[0166] If we now choose the load impedance R_l to optimize the efficiency by η_P(R_l)=0, we get the optimal load impedance

[0167] $\begin{array}{cc}\frac{R_{l^{*}}}{R_d}=\sqrt{1+\frac{{\left(\omega M\right)}^2}{R_s{R}_d}}& \left(32\right)\end{array}$
and the maximum possible efficiency

[0168] $\begin{array}{cc}{\eta }_{P^{*}}=\frac{R_{l^{*}}/R_d-1}{R_{l^{*}}/R_d+1}={\left[\frac{\omega M/\sqrt{R_s{R}_d}}{1+\sqrt{1+{\left(\omega M/\sqrt{R_s{R}_d}\right)}^2}}\right]}^2.& \left(33\right)\end{array}$

[0169] To check now the correspondence with the CMT model, note that κ_l=R_l/2L_d, Γ_d=R_d/2L_d, Γ_s=R_s/2L_s, and κ=ωM/2√{square root over (L_sL_d)}, so then U_l=κ_l/Γ_d=R_l/R_d and U=κ/√{square root over (Γ_sΓ_d)}=ωM/√{square root over (R_sR_d)}. Therefore, the condition Eq. (32) is identical to the condition Eq. (14) and the optimal efficiency Eq. (33) is identical to the general Eq. (15). Indeed, as the CMT analysis predicted, to get a large efficiency, we need to design a system that has a large strong-coupling factor U.

2.2.2 Optimization of U

[0170] The results above can be used to increase or optimize the performance of a wireless energy transfer system, which employs capacitively-loaded coils. For example, from the scaling of Eq. (27) with the different system parameters, one sees that to maximize the system figure-of-merit U, in some examples, one can:

• • [0171] Decrease the resistivity of the conducting material. This can be achieved, for example, by using good conductors (such as copper or silver) and/or lowering the temperature. At very low temperatures one could use also superconducting materials to achieve extremely good performance.
  • [0172] Increase the wire radius a. In typical examples, this action can be limited by physical size considerations. The purpose of this action is mainly to reduce the resistive losses in the wire by increasing the cross-sectional area through which the electric current is flowing, so one could alternatively use also a Litz wire or a ribbon instead of a circular wire.
  • [0173] For fixed desired distance D of energy transfer, increase the radius of the loop r. In typical examples, this action can be limited by physical size considerations, typically especially for the device.
  • [0174] For fixed desired distance vs. loop-size ratio D/r, decrease the radius of the loop r. In typical examples, this action can be limited by physical size considerations.
  • [0175] Increase the number of turns N. (Even though Eq. (27) is expected to be less accurate for N>1, qualitatively it still provides a good indication that we expect an improvement in the coupling-to-loss ratio with increased N.) In typical examples, this action can be limited by physical size and possible voltage considerations, as will be discussed in following paragraphs.
  • [0176] Adjust the alignment and orientation between the two coils. The figure-of-merit is optimized when both cylindrical coils have exactly the same axis of cylindrical symmetry (namely they are “facing” each other). In some examples, particular mutual coil angles and orientations that lead to zero mutual inductance (such as the orientation where the axes of the two coils are perpendicular and the centers of the two coils are on one of the two axes) should be avoided.
  • [0177] Finally, note that the height of the coil h is another available design parameter, which can have an impact to the performance similar to that of its radius r, and thus the design rules can be similar.

[0178] The above analysis technique can be used to design systems with desired parameters. For example, as listed below, the above described techniques can be used to determine the cross sectional radius a of the wire which one should use when designing as system two same single-turn loops with a given radius in order to achieve a specific performance in terms of U=κ/Γ at a given D/r between them, when the material is copper (σ=5.998·10⁷ S/m):
D/r=5, U≧10, r=30 cma≧9 mm
D/r=5, U≧10, r=5 cma≧3.7 mm
D/r=5, U≧20, r=30 cma≧20 mm
D/r=5, U≧20, r=5 cma≧8.3 mm
D/r=10, U≧1, r=30 cma≧7 mm
D/r=10, U≧1, r=5 cma≧2.8 mm
D/r=10, U≧3, r=30 cma≧25 mm
D/r=10, U≧3, r=5 cma≧10 mm

[0179] Similar analysis can be done for the case of two dissimilar loops. For example, in some examples, the device under consideration is very specific (e.g. a laptop or a cell phone), so the dimensions of the device object (r_d, h_d, a_d, N_d) are very restricted. However, in some such examples, the restrictions on the source object (r_s, h_s, a_s, N_s) are much less, since the source can, for example, be placed under the floor or on the ceiling. In such cases, the desired distance is often well defined, based on the application (e.g. D˜1 m for charging a laptop on a table wirelessly from the floor). Listed below are examples (simplified to the case N_s=N_d=1 and h_s=h_d=0) of how one can vary the dimensions of the source object to achieve the desired system performance in terms of U_sd=κ/√{square root over (Γ_sΓ_d)}, when the material is again copper (σ=5.998·10⁷ S/m):
D=1.5 m, U_sd≧15, r_d=30 cm, a_d=6 mmr_s=1.158 m, a_s≧5 mm
D=1.5 m, U_sd≧30, r_d=30 cm, a_d=6 mmr_s=1.15 m, a_s≧33 mm
D=1.5 m, U_sd≧1, r_d=5 cm, a_d=4 mmr_s=1.119 m, a_s≧7 mm
D=1.5 m, U_sd≧2, r_d=5 cm, a_d=4 mmr_s=1.119 m, a_s≧52 mm
D=2 m, U_sd≧10, r_d=30 cm, a_d=6 mmr_s=1.518 m, a_s≧7 mm
D=2 m, U_sd≧20, r_d=30 cm, a_d=6 mmr_s=1.514 m, a_s≧50 mm
D=2 m, U_sd≧0.5, r_d=5 cm, a_d=4 mmr_s=1.491 m, a_s≧5 mm
D=2 m, U_sd≧1, r_d=5 cm, a_d=4 mmr_s=1.491 m, a_s≧36 mm
2.2.3 Optimization of k

[0180] As described below, in some examples, the quality factor Q of the resonant objects is limited from external perturbations and thus varying the coil parameters cannot lead to improvement in Q. In such cases, one can opt to increase the strong-coupling factor U by increasing the coupling factor k. The coupling does not depend on the frequency and the number of turns. Therefore, in some examples, one can:

• • [0181] Increase the wire radii a₁ and a₂. In typical examples, this action can be limited by physical size considerations.
  • [0182] For fixed desired distance D of energy transfer, increase the radii of the coils r₁ and r₂. In typical examples, this action can be limited by physical size considerations, typically especially for the device.
  • [0183] For fixed desired distance vs. coil-sizes ratio D/√{square root over (r₁r₂)}, only the weak (logarithmic) dependence of the inductance remains, which suggests that one should decrease the radii of the coils r₁ and r₂. In typical examples, this action can be limited by physical size considerations.
  • [0184] Adjust the alignment and orientation between the two coils. In typical examples, the coupling is optimized when both cylindrical coils have exactly the same axis of cylindrical symmetry (namely they are “facing” each other). Particular mutual coil angles and orientations that lead to zero mutual inductance (such as the orientation where the axes of the two coils are perpendicular and the centers of the two coils are on one of the two axes) should obviously be avoided.
  • [0185] Finally, note that the heights of the coils h₁ and h₂ are other available design parameters, which can have an impact to the coupling similar to that of their radii r₁ and r₂, and thus the design rules can be similar.

[0186] Further practical considerations apart from efficiency, e.g. physical size limitations, will be discussed in detail below.

2.2.4 Optimization of Overall System Performance

[0187] In many cases, the dimensions of the resonant objects will be set by the particular application at hand. For example, when this application is powering a laptop or a cell-phone, the device resonant object cannot have dimensions larger than those of the laptop or cell-phone respectively. In particular, for a system of two loops of specified dimensions, in terms of loop radii r_s,d and wire radii a_s,d, the independent parameters left to adjust for the system optimization are: the number of turns N_s,d, the frequency f, the power-load consumption rate κ_l=R_l/2L_d and the power-generator feeding rate κ_g=R_g/2L_s, where R_g is the internal (characteristic) impedance of the generator.

[0188] In general, in various examples, the primary dependent variable that one wants to increase or optimize is the overall efficiency η. However, other important variables need to be taken into consideration upon system design. For example, in examples featuring capacitively-loaded coils, the design can be constrained by, for example, the currents flowing inside the wires I_s,d and the voltages across the capacitors V_s,d. These limitations can be important because for ˜Watt power applications the values for these parameters can be too large for the wires or the capacitors respectively to handle. Furthermore, the total loaded (by the load) quality factor of the device Q_d[l]=ω/2(Γ_d+Γ_l)=ωL_d/(R_d+R_d) and the total loaded (by the generator) quality factor of the source Q_s[g]=ω/2(Γ_s+Γ_g)=ωL_s/(R_s+R_g) are quantities that should be preferably small, because to match the source and device resonant frequencies to within their Q's, when those are very large, can be challenging experimentally and more sensitive to slight variations. Lastly, the radiated powers P_s,rad and P_d,rad should be minimized for concerns about far-field interference and safety, even though, in general, for a magnetic, non-radiative scheme they are already typically small. In the following, we examine then the effects of each one of the independent variables on the dependent ones.

[0189] We define a new variable wp to express the power-load consumption rate for some particular value of U through U_l=κ_l/Γ_d=√{square root over (1+wp·U)}. Then, in some examples, values which impact the choice of this rate are: U_l=1wp=0 to minimize the required energy stored in the source (and therefore I_s and V_s), U_l=√{square root over (1+U²)}>1wp=1 to maximize the efficiency, as seen earlier, or U_l>>1wp>>1 to decrease the required energy stored in the device (and therefore I_d and V_d) and to decrease or minimize Q_d[l]. Similar is the impact of the choice of the power-generator feeding rate U_g=κ_g/Γ_s, with the roles of the source/device and generator/load reversed.

[0190] In some examples, increasing N_s and N_d increases Q_s and Q_d, and thus U and the efficiency significantly, as seen before. It also decreases the currents I_s and I_d, because the inductance of the loops increases, and thus the energy W_s,d=L_s,d|I_s,d|²/2 required for given output power P_l can be achieved with smaller currents. However, in some examples, increasing N_d and thus Q_d can increase Q_d[l], P_d,rad and the voltage across the device capacitance V_d. Similar can be the impact of increasing N_s on Q_s[g], P_s,rad and V_s. As a conclusion, in some examples, the number of turns N_s and N_d should be chosen large enough (for high efficiency) but such that they allow for reasonable voltages, loaded Q's and/or powers radiated.

[0191] With respect to the resonant frequency, again, there is an optimal one for efficiency. Close to that optimal frequency Q_d[l] and/or Q_s[g] can be approximately maximum. In some examples, for lower frequencies the currents typically get larger but the voltages and radiated powers get smaller. In some examples, one should pick either the frequency that maximizes the efficiency or somewhat lower.

[0192] One way to decide on an operating regime for the system is based on a graphical method. Consider two loops of r_s=25 cm, r_d=15 cm, h_s=h_d=0, a_s=a_d=3 mm and distance D=2 m between them. In FIG. 12, we plot some of the above dependent variables (currents, voltages and radiated powers normalized to 1 Watt of output power) in terms of frequency f and N_d, given some choice for wp and N_s. FIG. 12 depicts the trend of system performance explained above. In FIG. 13, we make a contour plot of the dependent variables as functions of both frequency and wp but for both N_s and N_d fixed. For example, a reasonable choice of parameters for the system of two loops with the dimensions given above are: N_s=2, N_d=6, f=10 MHz and wp=10, which gives the following performance characteristics: η=20.6%, Q_d[l]=1264, I_s=7.2 A, I_d=1.4 A, V_s=2.55 kV, V_d=2.30 kV, P_s,rad=0.15 W, P_d,rad=0.006 W. Note that the results in FIGS. 12, 13 and the just above calculated performance characteristics are made using the analytical formulas provided above, so they are expected to be less accurate for large values of N_s, N_d, but still they give a good estimate of the scalings and the orders of magnitude.

[0193] Finally, one could additionally optimize for the source dimensions, since usually only the device dimensions are limited, as discussed earlier. Namely, one can add r_s and a_s in the set of independent variables and optimize with respect to these too for all the dependent variables of the problem (we saw how to do this only for efficiency earlier). Such an optimization would lead to improved results.

[0194] In this description, we propose that, if one ensures operation in the strongly-coupled regime at mid-range distances, at least medium-power transmission (˜W) at mid-range distances with high efficiency is possible.

2.3 Inductively-Loaded Conducting Rods

[0195] A straight conducting rod of length 2h and cross-sectional radius a has distributed capacitance and distributed inductance, and therefore it supports a resonant mode of angular frequency ω. Using the same procedure as in the case of self-resonant coils, one can define an effective total inductance L and an effective total capacitance C of the rod through formulas Eqs. (19) and (20). With these definitions, the resonant angular frequency and the effective impedance are given again by the common formulas ω=1/√{square root over (LC)} and Z=√{square root over (L/C)} respectively. To calculate the total inductance and capacitance, one can assume again a sinusoidal current profile along the length of the conducting wire. When interested in the lowest mode, if we denote by x the coordinate along the conductor, such that it runs from −h to +h, then the current amplitude profile would have the form I(x)=I_o cos(πx/2h), since it has to be zero at the open ends of the rod. This is the well-known half-wavelength electric dipole resonant mode.

[0196] In some examples, one or more of the resonant objects are inductively-loaded conducting rods. Referring to FIG. 14, a straight conducting rod of length 2h and cross-sectional radius a, as in the previous paragraph, is cut into two equal pieces of length h, which are connected via a coil wrapped around a magnetic material of relative permeability μ, and everything is surrounded by air. The coil has an inductance L_c, which is added to the distributed inductance of the rod and thus modifies its resonance. Note however, that the presence of the center-loading inductor modifies significantly the current distribution inside the wire and therefore the total effective inductance L and total effective capacitance C of the rod are different respectively from L_s and C_s, which are calculated for a self-resonant rod of the same total length using a sinusoidal current profile, as in the previous paragraph. Since some current is running inside the coil of the external loading inductor, the current distribution j inside the rod is reduced, so L<L_s, and thus, from the charge conservation equation, the linear charge distribution ρ_l flattens out towards the center (being positive in one side of the rod and negative in the other side of the rod, changing abruptly through the inductor), so C>C_s. The resonant frequency for this system is ω=1/√{square root over ((L+L_c)C)}<ω_s=1/√{square root over (L_sC_s)}, and I(x)→I_o cos(πx/2h) L→L_s ω→ω_s, as L_c→0.

[0197] In general, the desired CMT parameters can be found for this system, but again a very complicated solution of Maxwell's Equations is generally required. In a special case, a reasonable estimate for the current distribution can be made. When L_c>>L_s>L, then ω≈1/√{square root over (L_cC)}<<ω_s and Z≈√{square root over (L_cC)}>>Z_s, while the current distribution is triangular along the rod (with maximum at the center-loading inductor and zero at the ends) and thus the charge distribution is positive constant on one half of the rod and equally negative constant on the other side of the rod. This allows us to compute numerically C from Eq. (20). In this case, the integral in Eq. (20) can actually be computed analytically, giving the formula 1/C=1/(π∈_oh)[ln(h/a)−1]. Explicit analytical formulas are again available for R from Eq. (21) and (22), since I_rms=I_o, |p|=q_oh and |m|=0 (namely only the electric-dipole term is contributing to radiation), so we can determine also Q_abs=1/ωCR_abs and Q_rad=1/ωCR_rad. At the end of the calculations, the validity of the assumption of triangular current profile is confirmed by checking that indeed the condition L_c>>L_sω<<ω_s is satisfied. This condition is relatively easily satisfied, since typically a conducting rod has very small self-inductance L_s to begin with.

[0198] Another important loss factor in this case is the resistive loss inside the coil of the external loading inductor L_c and it depends on the particular design of the inductor. In some examples, the inductor is made of a Brooks coil, which is the coil geometry which, for fixed wire length, demonstrates the highest inductance and thus quality factor. The Brooks coil geometry has N_Bc turns of conducting wire of cross-sectional radius a_Bc wrapped around a cylindrically symmetric coil former, which forms a coil with a square cross-section of side r_Bc, where the inner side of the square is also at radius r_Bc (and thus the outer side of the square is at radius 2r_Bc), therefore N_Bc≈(r_Bc/2a_Bc)². The inductance of the coil is then L_c=2.0285 μ_or_BcN_Bc²≈2.0285 μ_or_Bc⁵/8a_Bc⁴ and its resistance

[0199] $R_C\approx \frac{1}{\sigma }\frac{l_{\mathrm{Bc}}}{\pi a_{\mathrm{Bc}}^2}\sqrt{1+\frac{{\mu }_o\mathrm{\omega \sigma }}{2}{\left(\frac{a_{\mathrm{Bc}}}{2}\right)}^2},$
where the total wire length is l_Bc≈2π(3r_Bc/2)N_Bc≈3πr_Bc³/4a_Bc² and we have used an approximate square-root law for the transition of the resistance from the dc to the ac limit as the skin depth varies with frequency.

[0200] The external loading inductance L_c provides the freedom to tune the resonant frequency. For example, for a Brooks coil with a fixed size r_Bc, the resonant frequency can be reduced by increasing the number of turns N_Bc by decreasing the wire cross-sectional radius a_Bc. Then the desired resonant angular frequency ω=1/√{square root over (L_cC)} is achieved for a_Bc≈(2.0285 μ_or_Bc⁵ω²C)^1/4 and the resulting coil quality factor is Q_c≈0.169 μ_oσr_Bc²ω/√{square root over (1+ω²μ_oσ√{square root over (2.0285μ_o(r_Bc/4)⁵C)})}. Then, for the particular simple case L_c>>L_s, for which we have analytical formulas, the total Q=1/ωC(R_c+R_abs+R_rad) becomes highest at some optimal frequency ω_Q, reaching the value Q_max, both determined by the loading-inductor specific design. For example, for the Brooks-coil procedure described above, at the optimal frequency Q_max≈Q_c≈0.8(μ_oσ²r_Bc³/C)^1/4. At lower frequencies it is dominated by ohmic loss inside the inductor coil and at higher frequencies by radiation. Note, again, that the above formulas are accurate as long as ω_Q<<ω_s and, as explained above, this is easy to design for by using a large inductance.

[0201] The results of the above analysis for two examples, using Brooks coils, of subwavelength modes of λ/h≧200 (namely highly suitable for near-field coupling and well within the quasi-static limit) at the optimal frequency ω_Q are presented in Table 5.

[0202] Table 5 shows in parentheses (for similarity to previous tables) analytical results for the wavelength and absorption, radiation and total loss rates, for two different cases of subwavelength-loop resonant modes. Note that for conducting material copper (σ=5.998·10⁷ S/m) was used. The results show that, in some examples, the optimal frequency is in the low-MHz microwave range and the expected quality factors are Q_abs≧1000 and Q_rad≧100000.

[0203]

TABLE 5
single rod	λ/h	f (MHz)	Qrad	Qabs	Q
h = 30 cm,	(403.8)	(2.477)	(2.72 * 106)	(7400)	(7380)
a = 2 cm
μ = 1,
rBc = 2 cm,
aBc = 0.88 mm,
NBc = 129
h = 10 cm,	(214.2)	(14.010)	(6.92 * 105)	(3908)	(3886)
a = 2 mm
μ = 1,
rBc = 5 mm,
aBc = 0.25 mm,
NBc = 103

[0204] In some examples, energy is transferred between two inductively-loaded rods. For the rate of energy transfer between two inductively-loaded rods 1 and 2 at distance D between their centers, the mutual capacitance M_C can be evaluated numerically from Eq. (23) by using triangular current distributions in the case ω<<ω_s. In this case, the coupling is only electric and again we have an analytical formula, which, in the quasi-static limit h<<D<<λ and for the relative orientation such that the two rods are aligned on the same axis, is 1/M_C≈1/2π∈_o·(h₁h₂)²/D³ which means that k ∝(√{square root over (h₁h₂)}/D)³ is independent of the frequency ω. One can then get the resultant strong-coupling factor U.

[0205] It can be seen that the optimal frequency ω_U, where the figure-of-merit is maximized to the value U_max, is close to the frequency ω_Q1Q2, where Q₁Q₂ is maximized, since k does not depend much on frequency (at least for the distances D<<λ of interest for which the quasi-static approximation is still valid). Therefore, the optimal frequency ω_U≈ω_Q1Q2 is mostly independent of the distance D between the two rods and lies between the two frequencies ω_Q1 and ω_Q2 at which the single-rod Q₁ and Q₂ respectively peak. In some typical examples, one can tune the inductively-loaded conducting rods, so that their angular eigenfrequencies are close to ω_U within Γ_U, which is half the angular frequency width for which U>U_max/2.

[0206] Referring to Table 6, in parentheses (for similarity to previous tables) analytical results based on the above are shown for two systems each composed of a matched pair of the loaded rods described in Table 5. The average wavelength and loss rates are shown along with the coupling rate and coupling to loss ratio figure-of-merit U=κ/Γ as a function of the coupling distance D, for the two cases. Note that for Γ_rad the single-rod value is used. Again we chose L_c>>L_s to make the triangular-current assumption a good one and computed M_C numerically from Eq. (23). The results show that for medium distances D/h=10−3 the expected coupling-to-loss ratios are in the range U˜0.5-100.

[0207]

TABLE 6
pair of rods	D/h	Qκ = 1/k	U
h = 30 cm, a = 2 cm	3	(70.3)	(105.0)
μ = 1, rBc = 2 cm, aBc = 0.88 mm,	5	(389)	(19.0)
NBc = 129
λ/h ≈ 404	7	(1115)	(6.62)
Q ≈ 7380	10	(3321)	(2.22)
h = 10 cm, a = 2 mm	3	(120)	(32.4)
μ = 1, rBc = 5 mm, aBc = 0.25 mm,	5	(664)	(5.85)
NBc = 103
λ/h ≈ 214	7	(1900)	(2.05)
Q ≈ 3886	10	(5656)	(0.69)

2.4 Dielectric Disks

[0208] In some examples, one or more of the resonant objects are dielectric objects, such as disks. Consider a two dimensional dielectric disk object, as shown in FIG. 15(a), of radius r and relative permittivity c surrounded by air that supports high-Q “whispering-gallery” resonant modes. The loss mechanisms for the energy stored inside such a resonant system are radiation into free space and absorption inside the disk material. High-Q_rad and long-tailed subwavelength resonances can be achieved when the dielectric permittivity ∈ is large and the azimuthal field variations are slow (namely of small principal number m). Material absorption is related to the material loss tangent: Q_abs˜Re{∈}/Im {∈}. Mode-solving calculations for this type of disk resonances were performed using two independent methods: numerically, 2D finite-difference frequency-domain (FDFD) simulations (which solve Maxwell's Equations in frequency domain exactly apart for spatial discretization) were conducted with a resolution of 30 pts/r; analytically, standard separation of variables (SV) in polar coordinates was used.

[0209]

TABLE 7
single disk	λ/r	Qabs	Qrad	Q
Re{ε} =	20.01 (20.00)	10103 (10075)	1988 (1992)	1661 (1663)
147.7,
m = 2
Re{ε} =	9.952 (9.950)	10098 (10087)	9078 (9168)	4780 (4802)
65.6,
m = 3

[0210] The results for two TE-polarized dielectric-disk subwavelength modes of λ/r≧10 are presented in Table 7. Table 7 shows numerical FDFD (and in parentheses analytical SV) results for the wavelength and absorption, radiation and total loss rates, for two different cases of subwavelength-disk resonant modes. Note that disk-material loss-tangent Im{∈}/Re{∈}=10⁻⁴ was used. (The specific parameters corresponding to the plot in FIG. 15(a) are highlighted with bold in the table.) The two methods have excellent agreement and imply that for a properly designed resonant low-loss-dielectric object values of Q_rad≧2000 and Q_abs˜10000 are achievable. Note that for the 3D case the computational complexity would be immensely increased, while the physics would not be significantly different. For example, a spherical object of ∈=147. 7 has a whispering gallery mode with m=2, Q_rad=13962, and λ/r=17.

[0211] The required values of ∈, shown in Table 7, might at first seem unrealistically large. However, not only are there in the microwave regime (appropriate for approximately meter-range coupling applications) many materials that have both reasonably high enough dielectric constants and low losses (e.g. Titania, Barium tetratitanate, Lithium tantalite etc.), but also E could signify instead the effective index of other known subwavelength surface-wave systems, such as surface modes on surfaces of metallic materials or plasmonic (metal-like, negative-∈) materials-or metallo-dielectric photonic crystals or plasmono-dielectric photonic crystals.

[0212] To calculate now the achievable rate of energy transfer between two disks 1 and 2, as shown in FIG. 15(b) we place them at distance D between their centers. Numerically, the FDFD mode-solver simulations give κ through the frequency splitting of the normal modes of the combined system (δ_E=2κ from Eq. (4)), which are even and odd superpositions of the initial single-disk modes; analytically, using the expressions for the separation-of-variables eigenfields E_1,2(r) CMT gives κ through
κ=ω₁/2·∫d³r∈₂(r)E₂*(r)E₁(r)/∫d³r∈(r)|E₁(r)|²,
where ∈_j (r) and ∈(r) are the dielectric functions that describe only the disk j (minus the constant ∈_o background) and the whole space respectively. Then, for medium distances D/r=10−3 and for non-radiative coupling such that D<2r_c, where r_c=mλ/2π is the radius of the radiation caustic, the two methods agree very well, and we finally find, as shown in Table 8, strong-coupling factors in the range U˜1-50. Thus, for the analyzed examples, the achieved figure-of-merit values are large enough to be useful for typical applications, as discussed below.

[0213]

TABLE 8
two disks	D/r	Qrad	Q = ω/2Γ	ω/2κ	κ/Γ
Re{ε} = 147.7,	3	2478	1989	46.9 (47.5)	42.4 (35.0)
m = 2	5	2411	1946	298.0 (298.0)	6.5 (5.6)
λ/r ≈ 20	7	2196	1804	769.7 (770.2)	2.3 (2.2)
Qabs ≈ 10093	10	2017	1681	1714 (1601)	0.98 (1.04)
Re{ε} = 65.6,	3	7972	4455	144 (140)	30.9 (34.3)
m = 3	5	9240	4824	2242 (2083)	2.2 (2.3)
λ/r ≈ 10	7	9187	4810	7485 (7417)	0.64 (0.65)
Qabs ≈ 10096

[0214] Note that even though particular examples are presented and analyzed above as examples of systems that use resonant electromagnetic coupling for wireless energy transfer, those of self-resonant conducting coils, capacitively-loaded resonant conducting coils, inductively-loaded resonant conducting rods and resonant dielectric disks, any system that supports an electromagnetic mode with its electromagnetic energy extending much further than its size can be used for transferring energy. For example, there can be many abstract geometries with distributed capacitances and inductances that support the desired kind of resonances. In some examples, the resonant structure can be a dielectric sphere. In any one of these geometries, one can choose certain parameters to increase and/or optimize U or, if the Q's are limited by external factors, to increase and/or optimize for k or, if other system performance parameters are of importance, to optimize those.

3 Coupled-Mode Theory for Prediction of Far-Field Radiation Interference

[0215] The two objects in an energy-transfer system generate radiation, which can sometimes be a significant part of the intrinsic losses, and can interfere in the far field. In the previous Sections, we analyzed systems, where this interference phenomenon was not in effect. In this description, we will repeat the analysis, including the interference effects and will show how it can be used to further enhance the power transmission efficiency and/or the radiated power.

[0216] The coupled-mode equations of Eqs. (1) fail to predict such an interference phenomenon. In fact, the inability to predict interference phenomena has often been considered inherent to coupled-mode theory (CMT). However, we show here that making only a simple extension to this model, it can actually very successfully predict such interference. The root of the problem stems from the fact that the coupling coefficients were tacitly assumed to be real. This is usually the case when dealing with proper (real) eigenmodes of a Hermitian (lossless) operator. However, this assumption fails when losses are included, as is for example the current case dealing with generally non-proper (leaky, radiative) eigenmodes of a non-Hermitian (lossy) operator. In this case, the coupling-matrix elements will generally be complex and their imaginary parts will be shown to be directly related to far-field radiation interference.

[0217] Imagine a system of many resonators in proximity to each other. When their resonances have close enough frequencies compared to their coupling rates, the CMT assumption is that the total-system field ψ is approximately determined only by these resonances as the superposition ψ(t)=Σ_na_n(t)ψ_n, where ψ_n is the eigenfield of the resonance n normalized to unity energy, and a_n is the field amplitude inside it, which corresponds, due to the normalization, to |a_n|² stored energy. The fundamental Coupled-Mode Equations (CME) of CMT are then those of the evolution of the vector a={a_n}

[0218] $\begin{array}{cc}\frac{d}{dt}a=-i\stackrel{\_}{\Omega }·a+i\stackrel{\_}{K}·a& \left(34\right)\end{array}$
where the frequency matrix Ω and the coupling matrix K are found usually using a Perturbation Theory (PT) approach.

[0219] We restate here one of the many perturbative formulations of CMT in a system of ElectroMagnetic (EM) resonators: Let μ=μ_o and ∈=∈_o+Σ_n∈_n be the magnetic-permeability and dielectric-permittivity functions of space that describe the whole system, where ∈_n is the permittivity of only the dielectric, reciprocal and generally anisotropic object n of volume V_n, in excess to the constant μ_o, ∈_o background space. Each resonator n, when alone in the background space, supports a resonant eigenmode of complex frequency Ω_n=ω_n−iΓ_n and field profiles ψ_n=[E_n, H_n] normalized to unity energy, satisfying the equations ∇×E_n=iΩ_nμ_oH_n and ∇×H_n=−iΩ_n(∈_o+∈_n)E_n, and the boundary condition {circumflex over (n)}×E_n=0 on the potential metallic surface S_n of object n. The whole system fields ψ=[E, H] satisfy the equations

[0220] $\nabla \times E=-\mu \frac{\partial }{\partial t}H\mathrm{and}\nabla \times H=\epsilon \frac{\partial }{\partial t}E,$
and the boundary condition {circumflex over (n)}×E=0 on S=Σ_nS_n. Then, start by expanding ∇·(E×H_n⁻−E_n⁻×H) and integrating over all space, apply the CMT superposition assumption, and finally use the PT argument that, when the coupling-rates between the resonators are small compared to their frequencies, within a sum only terms of lowest order in this small perturbation need to be kept. The result is the CME of Eq. (34), with Ω= W⁻¹· Ω_o· W, K= W⁻¹·K_o, where Ω_o=diag{Ω_n},

[0221] $\begin{array}{cc}{K}_{o,\mathrm{nm}}=\frac{{\Omega }_n}{4}{\int }_{V_m}dv\left(E_n^{-}·{\varepsilon }_m·E_m\right)+\frac{i}{4}\underset{S_m}{\oint }da\hat{n}·\left(E_n^{-}\times H_m\right)& \left(35\right)\\ W_{\mathrm{nm}}=\frac{1}{4}{\int }_{V_m}dv\left(E_n^{-}·\varepsilon ·E_m+H_n^{-}·\mu ·H_m\right)& \left(36\right)\end{array}$
and where ψⁿ=[E_n⁻, H_n⁻] satisfy the time-reversed equations (where Ω_n→−Ωn). The choice of these fields in the analysis rather than ψ_n*=[E_n*, H_n*] allows to treat also lossy (due to absorption and/or radiation) but reciprocal systems (so K is complex symmetric but non-Hermitian). In the limit, though, of weak loss (high-Q resonances), these two sets of fields can be approximately equal. Therefore, again to lowest order, W≈Ī, due to the unity-energy normalization, so Ω≈ Ω_o and for K the off-diagonal terms

[0222] $\begin{array}{cc}{K}_{\mathrm{nm}}\approx K_{o,\mathrm{mn}}\approx \frac{i}{4}{\int }_{V_m}dv\left(E_n^{*}·J_m\right);n\ne m& \left(37\right)\end{array}$
where J_m includes both the volume-polarization currents J_p,m=−iΩ_m∈_mE_m in V_m and the surface electric currents J_s,m={circumflex over (n)}×H_m on S_m, while the diagonal terms K_nn are higher-order small and can often lead to anomalous coupling-induced frequency shifts. The term of Eq. (37) can generally be complex K_nm=κ_nm+iΛ_nm and, even though the physical interpretation of its real part is well understood, as describing the coupling between the resonators, it is not so the case for its imaginary part

[0223] $\begin{array}{cc}\begin{array}{c}{\Lambda }_{\mathrm{nm}}=\frac{1}{4}\mathrm{Re}\left\{{\int }_{V_m}d{v\left[\mathrm{i\omega }{A}_n-\nabla {\varphi }_n\right]}^{*}·J_m\right\}\\ =\frac{1}{4}\mathrm{Re}\{{\int }_{V_m}d{v\left[{\int }_{V_n}dv\frac{e^{ikr_m-r_n}}{4\pi r_m-r_n}\left({\mathrm{i\omega \mu }}_0{J}_n+\frac{{\rho }_n}{{\varepsilon }_0}\nabla \right)\right]}^{*}·J_m\\ =\frac{\omega }{16\pi }{\int }_{V_m}dv{\int }_{V_n}dv\mathrm{Re}\left\{\left(\frac{{\rho }_n^{*}{\rho }_m}{{\varepsilon }_0}-{\mu }_0{J}_n^{*}·J_m\right)\frac{e^{ikr_m-r_n}}{4\pi r_m-r_n}\right\}\end{array}& \left(38\right)\end{array}$
where integration by parts was used for the ∇φ_n term and the continuity equation ∇·J=iωρ, with ρ being the volume charge density.

[0224] Towards understanding this term, let us consider two resonators 1, 2 and evaluate from Eqs. (34) the total power lost from the system

[0225] $\begin{array}{cc}\begin{array}{c}{P}_{\mathrm{loss}}=-\frac{d}{dt}\left({a_1}^2+{a_2}^2\right)\\ =2{\Gamma }_1{a_1}^2+2{\Gamma }_2{a_2}^2+4{\Lambda }_{12}\mathrm{Re}\left\{a_1^{*}{a}_2\right\}\end{array}& \left(39\right)\end{array}$

[0226] Clearly, the term involving an interaction between the two objects should not relate to material absorption, since this is a very localized process inside each object. We therefore split this lost power into absorbed and radiated in the following way
P_abs=2Γ_1,abs|a₁|²+2Γ_2,abs|a₂|²   (40)
P_rad=2Γ_1,rad|a₁|²+2Γ_2,rad|a₂|²+4Λ₁₂Re{a₁*a₂}  (41)
so Λ₁₂ is associated with the radiation from the two-object system. However, we have a tool to compute this radiated power separately: Antenna Theory (AT).

[0227] Let ζ_o=√{square root over (μ_o/∈_o)} and c_o=1/√{square root over (μ_o∈_o)} be the background impedance and light-velocity, and f=(g,f)=∫_V dv′J^v(r′)e^−ik·r′ the moment of the current-distribution 4-vector J^v=(c_oρ,J) of an electromagnetic resonator, where unity-energy normalization is again assumed for J^v and g={circumflex over (k)}·f, as can be shown using the continuity equation and integration by parts. The power radiated from one EM resonator is:

[0228] $\begin{array}{cc}{P}_{\mathrm{rad}}=2{\Gamma }_{\mathrm{rad}}{a}^2=\frac{{\varsigma }_o{k}^2}{32{\pi }^2}\left(\oint d\Omega {f}^2\right){a}^2& \left(42\right)\end{array}$
where |f|²=f*·f≡|f|²−|g|². The power radiated from an ‘array’ of two resonators 1 and 2, at vector-distance D between their centers, is given by:

[0229] $\begin{array}{cc}\begin{array}{c}{P}_{\mathrm{rad}}=\frac{{\varsigma }_o{k}^2}{32{\pi }^2}\oint d\Omega {a_1{f}_1+a_2{f}_2{e}^{-ik·D}}^2\\ =\frac{{\varsigma }_o{k}^2}{32{\pi }^2}\left[\begin{array}{c}\left(\oint d\Omega {f_1}^2\right){a_1}^2+\left(\oint d\Omega {f_2}^2\right){a_2}^2+\\ 2\mathrm{Re}\left\{\oint d\Omega f_1^{*}·f_2{e}^{-ik·D}{a}_1^{*}{a}_2\right\}\end{array}\right]\end{array}& \left(43\right)\end{array}$
where f₁*·f₂≡f₁*·f₂−g₁*·g₂. Thus, by comparing Eqs. (41) and (43), using Eq. (42),

[0230] $\begin{array}{cc}{\Lambda }_{12}=\frac{{ϛ}_o{k}^2}{64{\pi }^2}\frac{\mathrm{Re}\left\{\oint d\Omega f_1^{*}·f_2{e}^{-ik·D}{a}_1^{*}{a}_2\right\}}{\mathrm{Re}\left\{a_1^{*}{a}_2\right\}}.& \left(44\right)\end{array}$
namely Λ₁₂ is exactly the interference term in AT. By substituting for the 4-vector current-moments and making the change of variables r₁=r₁′, r₂=r₂′+D,

[0231] $\begin{array}{cc}\begin{array}{c}{\Lambda }_{12}=\frac{{ϛ}_o{k}^2}{64{\pi }^2}\frac{\mathrm{Re}\left\{{\int }_{V_1}^{}dv{\int }_{V_2}^{}d{\mathrm{vJ}}_1^{*}·J_2\oint d\Omega e^{-ik·\left(r_2-r_1\right)}{a}_1^{*}{a}_2\right\}}{\mathrm{Re}\left\{a_1^{*}{a}_2\right\}}\\ =\frac{{ϛ}_{o}k}{16\pi }\frac{\mathrm{Re}\left\{{\int }_{V_1}^{}dv{\int }_{V_2}^{}d{\mathrm{vJ}}_1^{*}·J_2\frac{\sin \left(kr_2-r_1\right)}{r_2-r_1}{a}_1^{*}{a}_2\right\}}{\mathrm{Re}\left\{a_1^{*}{a}_2\right\}}\end{array}& \left(45\right)\end{array}$
where we evaluated the integral over all angles of k with r₂−r₁.

[0232] Note now that Eqs. (38) and (45) will become identical, if we can take the currents J_1,2^v to be real. This is indeed the case for eigenmodes, where the field solution in bounded regions (such as those where the currents are flowing) is always stationary (in contrast to the leaky part of the eigenmode, which is radiative) and for high enough Q it can be chosen so that it is approximately real in the entire bounded region. Therefore, from either Eq. (38) or (45) we can write

[0233] $\begin{array}{cc}{\Lambda }_{12}=\frac{{ϛ}_{o}k}{16\pi }{\int }_{V_1}^{}dv{\int }_{V_2}^{}d{\mathrm{vJ}}_1·J_2\frac{\sin \left(kr_2-r_1\right)}{r_2-r_1}& \left(46\right)\end{array}$
and from Eq. (44), using Eq. (42), we can define the interference factor