CROSS-REFERENCE TO RELATED APPLICATIONS
This application is a divisional of U.S. application Ser. No. 15/180,425, filed Jun. 13, 2016, which claims the benefit of U.S. Provisional Application No. 62/247,955, filed Oct. 29, 2015, the disclosures of which are incorporated by reference.
BACKGROUND
Technical Field
The disclosure concerns the production of rotating microwaves in a plasma reactor chamber.
Background Discussion
In one approach for generating rotating microwaves in a plasma reactor chamber, microwaves are radiated into a cylindrical cavity from two ports separated spatially 90 degree. By setting a temporal phase difference between the microwaves from the two ports at 90 degrees, the TE_{111 }mode in a cylindrical cavity is rotated circularly with feedback control by two monitoring antennas, providing a plasma profile of high uniformity.
In another approach for generating rotating microwaves, temporal phases between the two microwaves radiated from the two ports are kept in phase. To produce rotation, an amplitude of the microwaves from one port is modulated in the form of A sin Ωt, while an amplitude of microwaves from the other port is modulated in the form of A cos Ωt. Here, Ω is an angular frequency of order of 1-1000 Hz, which is much smaller than that of a microwave carrier frequency of order of over 1 GHz. This dual injection rotates the TE_{111 }mode at a slow frequency Ω so as to slowly agitate a localized plasma, spreading the plasma into a wider area to further increase a uniformity of plasma distribution, particularly at high pressures.
However, the fast and slow rotations were provided only for the TE_{111 }mode. There is a need to provide such rotation for any mode, not just the TE_{111 }mode.
SUMMARY
In a plasma reactor comprising a cylindrical microwave cavity overlying a workpiece processing chamber, and first and second microwave input ports P and Q in a sidewall of the cylindrical microwave cavity spaced apart by an offset angle Δθ, a method is provided for generating rotating microwaves of mode TE_{mnl }or TM_{mnl }in the cylindrical microwave cavity, wherein n, m and l are user-selected values of a TE or TM mode. The method comprises: introducing into the cylindrical microwave cavity, through respective ones of the first and second coupling apertures, respective microwave signals separated by a temporal phase difference ΔØ; adjusting values of the offset angle Δθ and the temporal phase difference ΔØ to values which are a function of at least two of the user-selected TE or TM mode indices m, n and l so as to produce rotating microwaves of mode TE_{mnl }or TM_{mnl }in the cylindrical microwave cavity.
In one embodiment, the function is defined as:
$\hspace{1em}\{\begin{array}{cc}\frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{\Delta \theta}-\mathrm{\Delta \phi}}{2}=\frac{\pi}{2}+k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & k\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\\ \mathrm{\Delta \phi}=p\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & p\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\end{array}$
In one embodiment, the rotating microwaves rotate clockwise with the rotation frequency equal to an operational microwave frequency.
In one embodiment, to maximize the energy transfer efficiency of the clockwise rotation, the function is defined as:
$\hspace{1em}\{\begin{array}{cc}\frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{\Delta \theta}-\mathrm{\Delta \phi}}{2}=\frac{\pi}{2}+k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & k\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\\ \frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{\Delta \theta}+\mathrm{\Delta \phi}}{2}=p\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & p\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\end{array}$
In one embodiment, the function is defined as:
$\hspace{1em}\{\begin{array}{cc}\frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{\Delta \theta}+\mathrm{\Delta \phi}}{2}=\frac{\pi}{2}+k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & k\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\\ \mathrm{\Delta \phi}\ne p\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & p\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\end{array}$
In one embodiment, the rotating microwaves rotate anticlockwise with the rotation frequency equal to an operational microwave frequency.
In one embodiment, to maximize the energy transfer efficiency of the anticlockwise rotation, the function is defined as:
$\hspace{1em}\{\begin{array}{cc}\frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{\Delta \theta}+\mathrm{\Delta \phi}}{2}=\frac{\pi}{2}+k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & k\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\\ \frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{\Delta \theta}-\mathrm{\Delta \phi}}{2}=p\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & p\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\end{array}$
In one embodiment, a first one of the respective microwave signals is of a form:
H_{P }∝cos(η+mθ−ωt)+cos(η+mθ+ωt)
where ω is an angular frequency of the respective microwave signals and t is time, and
$\eta =0\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{or}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\frac{\pi}{2}.$
In one embodiment, a second one of the respective microwave signals is of a form:
H_{Q }∝cos[η+m(θ−Δθ)−(ωt−Δϕ)]+cos[η+m(θ−Δθ)+(ωt−Δϕ)]
where ω is an angular frequency of the microwave signals and t is time, and
$\eta =0\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{or}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\frac{\pi}{2}.$
In a plasma reactor comprising a cylindrical microwave cavity overlying a workpiece processing chamber, and first and second input ports P and Q in a sidewall of said cylindrical microwave cavity spaced apart by a general angle, a method is provided for generating rotating microwaves in said cylindrical microwave cavity with rotation frequency Ω_{a}, the method comprising:
setting said general angle to satisfy the following equations:
â=a
_{x }
{circumflex over (x)}+a
_{y }
ŷ
{circumflex over (b)}=b
_{x }
{circumflex over (x)}+b
_{y }
ŷ;
inputting to input ports P and Q microwave fields represented respectively by:
ζ_{Pa}=r cos(ωt+φ_{h})
ζ_{Qb}=s cos(ωt+φ_{h})
where r and s are defined in the following equations:
$r=\frac{\alpha \ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\Omega}_{a}\ue89et\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e{b}_{y}\mp \alpha \ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{sin}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\Omega}_{a}\ue89et\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e{b}_{x}}{V}$
$s=\frac{-\alpha \ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\Omega}_{a}\ue89et\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e{b}_{y}\pm \alpha \ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{sin}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\Omega}_{a}\ue89et\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e{a}_{x}}{V}$
and the sign ∓ in “r” determines whether the rotation is anticlockwise or clockwise.
In accordance with a further aspect, a plasma reactor comprises: a cylindrical microwave cavity overlying a workpiece processing chamber, and first and second input ports, P and Q, in a sidewall of the cylindrical microwave cavity spaced apart by an azimuthal angle; a microwave source having a microwave frequency and having a pair of microwave source outputs; a pair of respective waveguides, each of the respective waveguides having a microwave input end coupled to a respective one of the microwave source outputs and a microwave output end coupled to a respective one of the first and second input ports; a coupling aperture plate at the output end, and a rectangular coupling aperture in the coupling aperture plate; an iris plate between the coupling aperture plate and the microwave input end, and a rectangular iris opening in the iris plate.
In one embodiment, the rectangular coupling aperture and the rectangular iris opening have respective parallel axes along a long dimension of a respective one of the coupling aperture and the iris opening, the respective parallel axes being parallel to an axis of symmetry of the cylindrical microwave cavity.
In one embodiment, each of the waveguides has a microwave propagation direction between the microwave input end and the microwave output end, the microwave propagation direction extending toward an axis of symmetry of the cylindrical microwave cavity.
In one embodiment, the rectangular coupling aperture has long and short dimensions e and f, respectively, corresponding to a user-selected impedance.
In one embodiment, the rectangular iris opening has long and short dimensions c and d, respectively, corresponding to a user-selected resonance.
In one embodiment, the rectangular iris is a capacitive iris and has a long dimension parallel to an axis of symmetry of the cylindrical microwave cavity.
In one embodiment, the rectangular iris is an inductive iris and has a short dimension parallel to the axis of symmetry of the cylindrical microwave cavity.
BRIEF DESCRIPTION OF THE DRAWINGS
So that the manner in which the exemplary embodiments of the present invention are attained can be understood in detail, a more particular description of the invention, briefly summarized above, may be had by reference to the embodiments thereof which are illustrated in the appended drawings. It is to be appreciated that certain well known processes are not discussed herein in order to not obscure the invention.
FIG. 1A is cross-sectional elevational view of a plasma reactor that may be used in carrying out embodiments.
FIG. 1B is a plan view corresponding to FIG. 1A.
FIG. 1C is a plan view of a related reactor.
FIG. 2 is a schematic diagram of a system including the reactor of FIG. 1A.
FIG. 3 is a schematic diagram of another system including the reactor of FIG. 1A.
FIG. 3A is a block diagram depicting a method of operating the system of FIG. 3.
FIGS. 4 and 5 depict coordinate systems referred to in the detailed description.
FIG. 6 is a diagram of a system including a pair of power-feeding waveguides having impedance-shifting irises.
FIG. 6A is a plan view corresponding to FIG. 6.
FIGS. 7, 8 and 9 depict different irises for use in each waveguide of FIG. 6.
To facilitate understanding, identical reference numerals have been used, where possible, to designate identical elements that are common to the figures. It is contemplated that elements and features of one embodiment may be beneficially incorporated in other embodiments without further recitation. It is to be noted, however, that the appended drawings illustrate only exemplary embodiments of this invention and are therefore not to be considered limiting of its scope, for the invention may admit to other equally effective embodiments.
DETAILED DESCRIPTION
Introduction:
In the present description, microwave field rotations are provided for the general case of TE_{mnl }and TM_{mnl }in a cylindrical cavity, where m, n and l are suitable integers chosen by the user. Our recent experimental work confirms that a TE_{121 }mode makes a higher uniformity of plasma distribution under some conditions.
In addition, a method for changing a chamber impedance by using irises installed in a power-supplying waveguide is disclosed. In general, a cylindrical cavity has a bottom plate on which radiation slots are cut out to transfer microwave energy from the cavity to plasma. For a given design of the radiation slots, the chamber impedance is fixed. If the chamber impedance is in a region controlled by a stub tuner, the stub-tuner will make an impedance matching easily. Otherwise, the tuning becomes unpredictable or unstable to make an oscillation of the tuning position. Inversely, if the chamber impedance is controlled, it can be moved to tuner-preferred regions, which further leads to reduction of numbers of stubs, leading to cost reduction. The method proposed herein is simple, and moves the chamber impedance into wide ranges in the Smith chart as demonstrated in our recent experiments. The adoption of this method will provide stable plasma tuning and chamber-to-chamber etching/plasma matching.
Fast Rotation of TE_{mnl }and TM_{nml }Modes in a Cylindrical Cavity with a Microwave Carrier Frequency:
In this description, fast rotation is defined as a field rotation with the same rotation frequency as an operational microwave frequency. FIG. 1A is a simplified side view of a plasma reactor 100 including a processing chamber 110 enclosed by a wall 111 and containing gas under vacuum pressure and a workpiece support 112 for supporting a workpiece 114. A cylindrical cavity 120 overlying the processing chamber 110 is enclosed by a side wall 121a, a ceiling 121b and a floor 122 having slots 124 shown in FIG. 1B. The walls 121a and 111 can be connected by metal structures, depending upon application. A dielectric plate 130 provides a vacuum seal under the floor 122. The dielectric plate 130 is preferably formed of a material that is transparent to microwave radiation. FIG. 1C depicts an embodiment in which the floor 122 has an opening 810 and an auxiliary ignition electrode 820 is disposed in the opening 810 with a vacuum seal (not shown). The auxiliary ignition electrode 820 is driven by an RF source 830 of an RF frequency in a range of 100 Hz-10 MHz. The RF source 830 may include an impedance match (not illustrated). The floor 122 and/or the wall 111 of the processing chamber 110 can function as a ground plane with respect to the auxiliary ignition electrode 820. Alternatively, an auxiliary ignition electrode can be located on the wall 111 by providing an additional opening and vacuum seal. The electrode 820 and the ground plane are separated only by the opening 810. In summary, the auxiliary ignition electrode 820 together with the ground plane (i.e., the floor 122 and/or the wall 111 of the chamber 110) form a capacitively coupled RF igniting circuit to help ignition of plasma that is ultimately sustained by microwave power.
FIG. 2 depicts an embodiment in which first and second microwave input ports, P and Q, in the side wall 121a are located at azimuthal positions spaced apart by a non-orthogonal angle relative to one another. In FIG. 2, two identical microwave modules, Set-1 and Set-2, are connected to the cylindrical cavity 120 at input port P (where θ=0), and input port Q (where θ=θ_{q}), respectively. The other ends of the modules, Set-1 and Set-2, are connected to respective output signals, A_{1 }and A_{2}, of a dual digital seed (phase and amplitude) generator 340 that supplies microwave signals to the modules Set 1 and 2. In each module, the seed signal is amplified by a solid-state amplifier 350, which transmits it to a circulator 352 and a tuner 354, typically 3-pole stub tuner, to reduce reflection. The microwave is finally introduced into the cylindrical cavity 120 through a waveguide 360 with a radiating aperture, and excites eigen modes (resonances) in the cylindrical cavity 120. In general, transmission lines are used from the output of the amplifier 350 to the stub tuner 354. In this example, the radiating aperture is placed at the tip of the waveguide 360. A coaxial-to-waveguide transformer 358 is inserted between the tuner 354 and the waveguide 360. However, if a pole or loop antenna is adopted, the transformer 358 can be removed. In addition, a dummy load 362 is connected into one end of the circulator 352 to protect the amplifier 350.
Monitoring antennas 200a and 200b are orthogonally placed to receive microwave signals. The signal received by each one of the monitoring antennas 200a and 200b is processed by a signal feedback controller 340-1. In the feedback controller 340-1, the in- and quadrature-phase demodulation (IQ demodulation) is performed to measure the phase and amplitude of the received signal at the microwave frequency. When this phase and amplitude detection is performed for both the modules, Sets 1 and 2, the controller 340-1 calculates the mutual temporal phase difference Δφ and the amplitudes of the output signals, A_{1 }and A_{2 }using digital signal processing. Since the circularly fast rotation of TE_{mnl }and TM_{mnl }mode in a cylindrical cavity with a microwave carrier frequency requires Δφ=±90° and A_{1}=A_{2}, the controller 340-1 performs feedback-loop control, until the required relation is satisfied. This feed-back is operated independently from stub tuning works. Hence, as long as high speed controllers, such as an FPGA and a microcontroller, are used, a prompt conversion to the required condition is achieved in less than a millisecond.
Representation of Electromagnetic Fields of TE_{mnl }in a Resonant Cavity:
In FIG. 2, the angled (non-orthogonal) orientation of the input ports P and Q requires a new condition on a temporal phase delay Δφ of Q from that of P to make the circular fast rotation. As already stated, the feedback monitoring system can take care of control to make the perfect circular fast rotation. However, it is desirable to set a best initial value of Δφ so that feedback control time becomes minimized. Δφ is now derived for TE_{mnl }and TM_{mnl}. In the following, h=height of a cylindrical cavity, and R=radius of the cylindrical cavity.
For TE_{mnl}, the fields are represented for given a single integer ‘m’ in Gauss units:
$\begin{array}{cc}{\kappa}^{2}=\frac{{\omega}^{2}}{{c}^{2}}\ue89e\varepsilon -{k}_{\mathrm{zl}}^{2}\ue89e\text{}\ue89e{B}_{z}={J}_{m}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e{e}^{\mathrm{im}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta}\ue8a0\left({A}_{+}\ue89e{e}^{{\mathrm{ik}}_{\mathrm{zl}}\ue89ez}+{A}_{-}\ue89e{e}^{-{\mathrm{ik}}_{\mathrm{zl}}\ue89ez}\right)\ue89e\text{}\ue89e\overrightarrow{{B}_{t}}=\frac{{\mathrm{ik}}_{\mathrm{zl}}}{{\kappa}^{2}}\ue89e{e}^{\mathrm{im}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta}\ue8a0\left[\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{J}_{m}^{\prime}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\hat{r}+\frac{\mathrm{im}}{r}\ue89e{J}_{m}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\hat{\theta}\right]\ue89e\left({A}_{+}\ue89e{e}^{{\mathrm{ik}}_{\mathrm{zl}}\ue89ez}-{A}_{-}\ue89e{e}^{-{\mathrm{ik}}_{\mathrm{zl}}\ue89ez}\right)\ue89e\text{}\ue89e\overrightarrow{{E}_{t}}=\frac{i\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\omega}{c\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\kappa}^{2}}\ue89e{e}^{\mathrm{im}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta}\ue8a0\left[\frac{\mathrm{im}}{r}\ue89e{J}_{m}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\hat{r}-\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{J}_{m}^{\prime}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\hat{\theta}\right]\ue89e\left({A}_{+}\ue89e{e}^{{\mathrm{ik}}_{\mathrm{zl}}\ue89ez}+{A}_{-}\ue89e{e}^{-{\mathrm{ik}}_{\mathrm{zl}}\ue89ez}\right).& \left(1\right)\end{array}$
The boundary condition that the tangential components of electric fields in the cavity must vanish leads to the following relations:
$\begin{array}{cc}{A}_{+}=-{A}_{-}\equiv \frac{\mathrm{iA}}{2}\ue89e\text{}\ue89e{k}_{\mathrm{zl}}=\frac{l\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi}{h}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89el\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\ue89e\text{}\ue89e\kappa ={y}_{\mathrm{mn}}^{\prime}/R& \left(2\right)\end{array}$
where J′_{m}(y′_{mn})=0.
Then, the fields become
$\begin{array}{cc}{B}_{z}={\mathrm{AJ}}_{m}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e{e}^{\mathrm{im}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta}\ue89e\mathrm{sin}\ue8a0\left({k}_{\mathrm{zl}}\ue89ez\right)\ue89e\text{}\ue89e\overrightarrow{{B}_{t}}=A\ue89e\frac{{k}_{\mathrm{zl}}}{{\kappa}^{2}}\ue89e{e}^{\mathrm{im}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta}\ue8a0\left[\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{J}_{m}^{\prime}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\hat{r}+\frac{\mathrm{im}}{r}\ue89e{J}_{m}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\hat{\theta}\right]\ue89e\mathrm{cos}\ue8a0\left({k}_{\mathrm{zl}}\ue89ez\right)\ue89e\text{}\ue89e\overrightarrow{{E}_{t}}=A\ue89e\frac{\omega}{c\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\kappa}^{2}}\ue89e{e}^{\mathrm{im}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta}\ue8a0\left[\frac{-m}{r}\ue89e{J}_{m}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\hat{r}-i\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{J}_{m}^{\prime}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\hat{\theta}\right]\ue89e\mathrm{sin}\ue8a0\left({k}_{\mathrm{zl}}\ue89ez\right).& \left(3\right)\end{array}$
When considering two degenerate ‘n’ and ‘−n’ along with the temporal term e^{−iωt}, we can write the magnetic fields as:
$\begin{array}{cc}{B}_{z}={\mathrm{AJ}}_{m}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e{e}^{\mathrm{im}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta}\ue89e\mathrm{sin}\ue8a0\left({k}_{\mathrm{zl}}\ue89ez\right)\ue8a0\left[\alpha \ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{cos}\ue8a0\left(m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta -\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89et\right)+b\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{cos}\ue8a0\left(m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta +\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89et\right)\right]\ue89e\text{}\ue89e{B}_{r}=A\ue89e\frac{{k}_{\mathrm{zl}}}{\kappa}\ue89e{J}_{m}^{\prime}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\mathrm{cos}\ue8a0\left({k}_{\mathrm{zl}}\ue89ez\right)\ue8a0\left[a\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{cos}\ue8a0\left(m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta -\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89et\right)+b\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{cos}\ue8a0\left(m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta +\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89et\right)\right])\ue89e\text{}\ue89e{B}_{\theta}=-A\ue89e\frac{{\mathrm{mk}}_{\mathrm{zl}}}{{\kappa}^{2}\ue89er}\ue89e{J}_{m}^{\prime}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\mathrm{cos}\ue8a0\left({k}_{\mathrm{zl}}\ue89ez\right)\ue8a0\left[a\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{sin}\ue8a0\left(m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta -\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89et\right)+b\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{sin}\ue8a0\left(m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta +\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89et\right)\right]& \left(4\right)\end{array}$
where a and b are constants.
All the magnetic field components at a fixed (r,z) can be written with newly normalized constants a and b in the form of
B=a cos(η+mθ−ωt)+b cos(η+mθ+ωt) (5)
where
$\eta =0\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{or}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\frac{\pi}{2}.$
Specifically, in Eqn. (5), “a” and “b” are amplitude coefficients of the anticlockWlSE and clockwise rotation, respectively.
Representation of electromagnetic fields of TM_{mnl }in a resonant cavity:
For TM_{mnl}, the fields are represented for given a single integer, ‘m’ in Gauss units:
$\begin{array}{cc}{\kappa}^{2}=\frac{{\omega}^{2}}{{c}^{2}}\ue89e\varepsilon -{k}_{\mathrm{zl}}^{2}\ue89e\text{}\ue89e{E}_{z}={J}_{m}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e{e}^{\mathrm{im}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta}\ue8a0\left({A}_{+}\ue89e{e}^{{\mathrm{ik}}_{\mathrm{zl}}\ue89ez}+{A}_{-}\ue89e{e}^{-{\mathrm{ik}}_{\mathrm{zl}}\ue89ez}\right)\ue89e\text{}\ue89e\overrightarrow{{E}_{t}}=\frac{{\mathrm{ik}}_{\mathrm{zl}}}{{\kappa}^{2}}\ue89e{e}^{\mathrm{im}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta}\ue8a0\left[\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{J}_{m}^{\prime}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\hat{r}+\frac{\mathrm{im}}{r}\ue89e{J}_{m}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\hat{\theta}\right]\ue89e\left({A}_{+}\ue89e{e}^{{\mathrm{ik}}_{\mathrm{zl}}\ue89ez}-{A}_{-}\ue89e{e}^{-{\mathrm{ik}}_{\mathrm{zl}}\ue89ez}\right)\ue89e\text{}\ue89e\overrightarrow{{B}_{t}}=-\frac{i\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{\omega \varepsilon}}{c\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\kappa}^{2}}\ue89e{e}^{\mathrm{im}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta}\ue8a0\left[\frac{\mathrm{im}}{r}\ue89e{J}_{m}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\hat{r}-\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{J}_{m}^{\prime}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\hat{\theta}\right]\ue89e\left({A}_{+}\ue89e{e}^{{\mathrm{ik}}_{\mathrm{zl}}\ue89ez}+{A}_{-}\ue89e{e}^{-{\mathrm{ik}}_{\mathrm{zl}}\ue89ez}\right).& \left(6\right)\end{array}$
In a similar manner to TE_{mnl}, the boundary condition that the tangential components of electric fields in the cavity must vanish leads to the following relations with slight changes
$\begin{array}{cc}{A}_{+}={A}_{-}\equiv \frac{\mathrm{iA}}{2}\ue89e\text{}\ue89e{k}_{\mathrm{zl}}=\frac{l\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi}{h}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89el\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\ue89e\text{}\ue89e\kappa ={y}_{\mathrm{mn}}/R& \left(7\right)\end{array}$
where J_{m}(y_{mn})=0.
$\begin{array}{cc}\begin{array}{c}{E}_{z}={\mathrm{AJ}}_{m}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e{\u212f}^{\mathrm{im}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta}\ue89e\mathrm{cos}\ue8a0\left({k}_{\mathrm{zl}}\ue89ez\right)\\ {\overrightarrow{E}}_{t}=-A\ue89e\frac{{k}_{\mathrm{zl}}}{{\kappa}^{2}}\ue89e{\u212f}^{\mathrm{im}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta}[\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{J}_{m}^{\prime}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\hat{r}\ue89e\phantom{\rule{0.6em}{0.6ex}}+\frac{\mathrm{im}}{r}\ue89e{J}_{m}\ue8a0\left(\kappa \ue89er\right)\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\hat{\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta \ue89e\phantom{\rule{0.3em}{0.3ex}}]}\ue89e\mathrm{sin}\ue8a0\left({k}_{\mathrm{zl}}\ue89ez\right)\\ {\overrightarrow{B}}_{t}=A\ue89e\frac{\mathrm{\omega \varepsilon}}{{c\ue89e\kappa}^{2}}\ue89e{\u212f}^{\mathrm{im}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta}[\frac{m}{r}\ue89e{J}_{m}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\hat{r}\ue89e\phantom{\rule{0.6em}{0.6ex}}+i\ue89e\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{J}_{m}^{\prime}\ue8a0\left(\kappa \ue89er\right)\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\hat{\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta \ue89e\phantom{\rule{0.3em}{0.3ex}}]}\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{cos}\ue8a0\left({k}_{\mathrm{zl}}\ue89ez\right).\end{array}& \left(8\right)\end{array}$
When considering both n and −n along with the temporal term e^{−iωt}, we can write the magnetic fields as
$\begin{array}{cc}\hspace{0.17em}\begin{array}{c}{B}_{z}=0\\ {B}_{r}=-A\ue89e\frac{\mathrm{\omega \varepsilon}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89em}{{c\ue89e\kappa}^{2}\ue89er}\ue89e{J}_{m}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{cos}\ue8a0\left({k}_{\mathrm{zl}}\ue89ez\right)\ue8a0\left[a\ue89e\mathrm{cos}\ue8a0\left(m\ue89e\theta -\omega \ue89et\right)+b\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{cos}\ue8a0\left(m\ue89e\theta +\omega \ue89et\right)\right]\\ {B}_{\theta}=A\ue89e\frac{\mathrm{\omega \varepsilon}}{c\ue89e\kappa \ue89er}\ue89e{J}_{m}^{\prime}\ue8a0\left(\kappa \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89er\right)\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{cos}\ue8a0\left({k}_{\mathrm{zl}}\ue89ez\right)\ue8a0\left[a\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{sin}\ue8a0\left(m\ue89e\theta -\omega \ue89et\right)+b\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{sin}\ue8a0\left(m\ue89e\theta +\omega \ue89et\right)\right].\end{array}& \left(9\right)\end{array}$
All the magnetic field components at a fixed (r,z) can be written with newly normalized constants a and b in the form of:
B=a cos(η+mθ−ωt)+b cos(η+mθ+ωt) (10)
where
$\eta =0\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{or}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\frac{\pi}{2}.$
Since Eqn. (10) is of identical form to Eqn. (5), the following discussions can be applied to both TE_{mnl }and TM_{mnl}. For the sake of brevity, the term η in Eqns. (5) and (10) will be dropped in the following discussion.
Single and Dual Injection for TE_{mnl }and TM_{mnl}:
When considering wave excitation from Port P, anticlockwise and clockwise rotations are excited with equal probabilities as a first approximation. Then, the excited wave can be written by renormalizing the coefficients a and b in Eqn. (10) as unity:
H_{P}=cos(mθ−ωt)+cos(mθ+ωt). (11)
Next, when exciting a wave from Port Q with the same power and frequency, however, with a temporal phase delay of Δϕ, the excited wave can be represented as:
H_{Q}=cos[m(θ−Δθ)−(ωt−Δϕ)]+cos[m(θ−Δθ)+(ωt−Δϕ)] (12)
where Δθ is the angular offset in position of Port Q relative to Port P, and Δϕ is the temporal phase difference between the microwave outputs A_{1 }and A_{2}. When exciting the cavity 120 from both input ports P and Q simultaneously, the excited wave can be given as a sum of Eqns. (11) and (12):
H_{tot}=cos(mθ−ωt)+cos(mθ+ωt)+cos[m(θ−Δθ)−(ωt−Δϕ)]+cos[m(θ−Δθ)+(ωt−Δϕ)].
Or, factoring the anticlockwise H_{+} and clockwise H_{−} components:
H_{tot}=H_{+}+H_{−} (13)
where
$\begin{array}{cc}{H}_{+}=\mathrm{cos}\ue8a0\left(m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta -\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89et\right)+\mathrm{cos}\ue8a0\left[m\ue8a0\left(\theta -\mathrm{\Delta \theta}\right)-\left(\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89et-\mathrm{\Delta \phi}\right)\right]=2\ue89e\mathrm{cos}\ue8a0\left[\frac{2\ue89em\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta -2\ue89e\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89et-m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta +\mathrm{\Delta \phi}}{2}\right]\ue89e\mathrm{cos}\ue8a0\left(\frac{\phantom{\rule{0.3em}{0.3ex}}\ue89em\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta -\mathrm{\Delta \phi}}{2}\right)& \left(14\right)\\ {H}_{-}=2\ue89e\mathrm{cos}\ue8a0\left(m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta +\omega \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89et+\frac{-m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta +\mathrm{\Delta \phi}}{2}\right)\ue89e\mathrm{cos}\ue8a0\left(\frac{\phantom{\rule{0.3em}{0.3ex}}\ue89em\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta +\mathrm{\Delta \phi}}{2}\right).& \left(15\right)\end{array}$
Condition for the Clockwise Rotation for TE_{mnl }and TM_{mnl}:
The anticlockwise term will vanish, if the last term of Eqn. (14) is null, explicitly:
$\begin{array}{cc}\frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta -\mathrm{\Delta \phi}}{2}=\frac{\pi}{2}+k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi \ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{for}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89ek=\mathrm{integer}.& \left(16\right)\end{array}$
If the following condition as well as that of Eqn. (16) are simultaneously satisfied,
$\begin{array}{cc}\frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta +\mathrm{\Delta \phi}}{2}=\frac{\pi}{2}+p\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi \ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{for}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89ep=\mathrm{integer}& \left(17\right)\end{array}$
then, neither the anticlockwise nor clockwise waves are excited. This simultaneous condition can be provided by:
$\hspace{1em}\{\begin{array}{cc}\frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta -\mathrm{\Delta \phi}}{2}=\frac{\pi}{2}+k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & k=\mathrm{integer}\\ \mathrm{\Delta \phi}=q\ue89e\pi & q=\mathrm{integer}\end{array}.$
Conversely, the necessary and sufficient condition to excite only the clockwise rotation for TE_{nml }or TM_{nml }can be summarized as:
$\begin{array}{cc}\{\begin{array}{cc}\frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta -\mathrm{\Delta \phi}}{2}=\frac{\pi}{2}+k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & k\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\\ \mathrm{\Delta \phi}\ne q\ue89e\pi & q\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\end{array}.& \left(18\right)\end{array}$
To maximize the energy transfer efficiency of the clockwise rotation, the last term of Eqn. (15) must be ±1, simultaneously with Eqn. (16), namely
$\begin{array}{cc}\{\begin{array}{cc}\frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta -\mathrm{\Delta \phi}}{2}=\frac{\pi}{2}+k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & k\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\\ \frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta +\mathrm{\Delta \phi}}{2}=p\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & p\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\end{array}& \phantom{\rule{0.3em}{0.3ex}}\end{array}$
which can be reduced to
$\begin{array}{cc}\mathrm{\Delta \phi}=-\frac{\pi}{2}+\left(p-k\right)\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi \ue89e\phantom{\rule{1.7em}{1.7ex}}\ue89e\mathrm{\Delta \theta}=\frac{1+2\ue89e\left(k+p\right)}{2\ue89em}\ue89e\pi \ue89e\phantom{\rule{1.7em}{1.7ex}}\ue89ek,p\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integers}.& \left(19\right)\end{array}$
Eqn. (19) is included as a special case of Eqn. (18). However, Eqn. (19) is preferable because of its maximum efficiency. A further simplification is given by setting k=p
$\begin{array}{cc}\mathrm{\Delta \phi}=-\frac{\pi}{2}\ue89e\phantom{\rule{1.7em}{1.7ex}}\ue89e\mathrm{\Delta \theta}=\frac{1+4\ue89e\kappa}{2\ue89em}\ue89e\pi \ue89e\phantom{\rule{1.7em}{1.7ex}}\ue89ek\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}.& \left(20\right)\end{array}$
Microwave dual injections to excite a purely clockwise rotation with the maximum efficiency are summarized as follows:
Case of TE_{111}:
$\begin{array}{cc}\mathrm{Port}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89eQ\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{separated}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{from}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{Port}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89eP\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{by}\ue89e\phantom{\rule{0.8em}{0.8ex}}\pm \frac{\pi}{2}\ue89e\text{}\ue89e\mathrm{Temporal}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{phase}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{delay}\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}-\frac{\pi}{2}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\left(i.e.\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{phase}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{advance}\right);& \left(21\right)\end{array}$
Case of TE_{121}:
$\begin{array}{cc}\mathrm{Port}\ue89e\phantom{\rule{1.1em}{1.1ex}}\ue89eQ\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{separated}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{from}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{Port}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89eP\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{by}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\frac{\pi}{4}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{or}\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\frac{5\ue89e\pi}{4}\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\text{}\ue89e\mathrm{Temporal}\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{phase}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{delay}\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}-\frac{\pi}{2}\ue89e\phantom{\rule{1.1em}{1.1ex}}\ue89e\left(i.e.\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{phase}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{advance}\right).& \left(22\right)\end{array}$
Condition for Anticlockwise Rotation for TE_{mnl }and TM_{mnl}:
In the same manner, the necessary and sufficient condition to excite only the anticlockwise rotation for TE_{mnl }or TM_{mnl }can be summarized as:
$\begin{array}{cc}\{\begin{array}{cc}\frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta +\mathrm{\Delta \phi}}{2}=\frac{\pi}{2}+k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & k\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\\ \mathrm{\Delta \phi}\ne q\ue89e\pi & q\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\end{array}.& \left(23\right)\end{array}$
Eqn. (23) defines Δθ and ΔØ as a function of the user-selected indices m, n and l of the modes TE_{mnl }or TM_{mnl}. To maximize the energy efficiency of the anticlockwise rotation, the last term of Eqn. (16) should be ±1, simultaneously with Eqn. (15), namely:
$\begin{array}{cc}\{\begin{array}{cc}\frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta +\mathrm{\Delta \phi}}{2}=\frac{\pi}{2}+k\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & k\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\\ \frac{m\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\Delta \ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta -\mathrm{\Delta \phi}}{2}=p\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\pi & p\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}\end{array}& \left(24\right)\end{array}$
which can be reduced to
$\begin{array}{cc}\mathrm{\Delta \phi}=\frac{\pi}{2}+\left(k-p\right)\ue89e\pi \ue89e\text{}\ue89e\mathrm{\Delta \theta}=\frac{1+2\ue89e\left(k+p\right)}{2\ue89em}\ue89e\pi \ue89e\text{}\ue89ek,p\ue89e\text{:}\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{integers}& \left(25\right)\end{array}$
Or, a simplification by setting k=p gives
$\begin{array}{cc}\mathrm{\Delta \phi}=\frac{\pi}{2}\ue89e\text{}\ue89e\mathrm{\Delta \theta}=\frac{1+4\ue89ek}{2\ue89em}\ue89e\pi \ue89e\text{}\ue89ek\ue89e\text{:}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{integer}.& \left(26\right)\end{array}$
Microwave dual injections to excite a purely clockwise rotation with the maximum efficiency are summarized as follows.
Case of TE_{111}:
$\begin{array}{cc}\mathrm{Port}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89eQ\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{seperated}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{from}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{Port}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89eP\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{by}\pm \frac{\pi}{2}\ue89e\text{}\ue89e\mathrm{Temporal}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{phase}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{delay}\ue89e\phantom{\rule{0.3em}{0.3ex}}=\frac{\pi}{2}& \left(27\right)\end{array}$
Case of TE_{121}:
$\begin{array}{cc}\mathrm{Port}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89eQ\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{seperated}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{from}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{Port}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89eP\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{by}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\frac{\pi}{4}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{or}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\frac{5\ue89e\pi}{4}\ue89e\text{}\ue89e\mathrm{Temporal}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{phase}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{delay}\ue89e\phantom{\rule{0.3em}{0.3ex}}=\frac{\pi}{2}.& \left(28\right)\end{array}$
Each one of Eqns. 18-20 and 23-26 defines Δθ and ΔØ as a function of the user-selected indices m, n and l of the modes TE_{mnl }or TM_{mnl}.
In summary, a rotating microwave is established in the cavity 120 for any resonant mode TE_{mnl }or TM_{mnl }of the cavity, where the user is free to choose the values of the mode indices n, m and l. This is accomplished by setting the temporal phase difference ΔØ and the azimuthal angle Δθ between the ports P and Q as functions of m, n and l, defined in an applicable one of the Eqns. 18-20 and 23-26. The foregoing is illustrated as a method in a block diagram of FIG. 3A. In FIG. 3A, a plasma reactor including a cylindrical microwave cavity overlying a workpiece processing chamber is provided as in FIG. 3. First and second input ports P and Q are provided in a sidewall of the cylindrical microwave cavity spaced apart by an offset angle Δθ, (block 600 of FIG. 3A). A next step is to generate rotating microwaves of mode TE_{mnl }or TM_{mnl }in the cylindrical microwave cavity, wherein at least two of m, n and l are user-selected values of a TE or TM mode (block 602). This is done by introducing into the cylindrical microwave cavity, through respective ones of the first and second coupling apertures, respective microwave signals separated by a temporal phase difference ΔØ (block 604 of FIG. 3A). The method includes adjusting values of the offset angle Δθ and the temporal phase difference ΔØ to values which are a function of at least two of the user-selected TE or TM mode indices m, n and l, so as to produce rotating microwaves of mode TE_{mnl }or TM_{mnl }in the cylindrical microwave cavity (block 606 of FIG. 3A).
Generalized Amplitude Modulation for a Slow Rotation of TE_{mnl }and TM_{mnl }Mode in a Cylindrical Cavity:
FIG. 3 depicts a modification of the embodiment of FIG. 2 for amplitude modulation for a slow rotation of TE_{mnl }and TM_{mnl }mode in a cylindrical cavity. It is the same as that of FIG. 2 except for the absence of monitoring antennas and absence of a signal feedback controller.
Amplitude Modulations Radiated from Ports P and Q:
Microwave fields radiated from Ports P and Q, where P and Q are spatially separated by 90 degrees, should have the following forms of amplitude modulation to make a slow rotation of frequency Ω_{a }on the order of 1-1000 Hz:
ζ_{Px}=α cos(ω_{a}t)cos(ωt+φ_{h}) (29)
ζ_{Qy}=±α sin(Ω_{a}t)cos(ωt+φ_{h}) (30)
where α is an arbitrary constant, Ω_{a }is an angular frequency of rotation, t is a time, and φ_{h }is an arbitrary initial phase, and the plus and minus signs of Eqn. (30) correspond to anticlockwise and clockwise rotations, respectively. Then, an excited wave in a cylindrical cavity can be represented by using an azimuthal angle θ:
η=2c cos(θ∓Ω_{a}t)cos(ωtφ_{h})=[2c cos(Ω_{a}t)cos θ+{±2c sin(Ω_{a}t)}sin θ]cos(ωt+φ_{h}) (31)
When rewriting Eqns. (29)-(30) in x-y coordinate system, it can be stated: a vector input
{right arrow over (ζ)}={α cos(Ω_{a}t){circumflex over (x)}±α sin(Ω_{a}t)ŷ}cos(ωt+φ_{h}) (32)
excites a vector wave of
{right arrow over (η)}={2c cos(Ω_{a}t){circumflex over (x)}±2c sin(Ω_{a}t)ŷ}cos(ωt+φ_{h}) (33)
where {circumflex over (x)} and ŷ are unit base vectors in x and y directions, respectively.
In FIG. 3, the Ports P and Q are not necessarily located at a 90 degree interval. However, it is required that an excited wave should have the form of Eqn. (33). This problem can be converted to a coordinate transformation from an orthogonal x-y system to an oblique a-b system as shown in FIG. 4.
In FIG. 4, a general vector {right arrow over (P)} is defined as
{right arrow over (P)}=p{circumflex over (x)}+qŷ=tâ+s{circumflex over (b)} (34)
where the base vectors in the a-b system are defined as
â=a_{x }{circumflex over (x)}+a_{y }ŷ (35)
{circumflex over (b)}=b_{x }{circumflex over (x)}+b_{y }ŷ (36)
Hence, when the ports P and Q are separated by 90 degrees, Eqn. (33) can be represented by
{right arrow over (P)}=α cos Ω_{a}t{circumflex over (x)}±α sin Ω_{a}t ŷ (36-2)
where the common temporal term cos(ωt+φ_{h}) has been skipped.
Thus, p and q in Eqn. (34) are defined as:
p=α cos Ω_{a}t (36-3)
q=±α sin Ω_{a}t (36-4)
To obtain the expression in the oblique system, let the reciprocal bases {circumflex over (α)} and {circumflex over (β)} correspond to the bases â and {circumflex over (b)}
$\begin{array}{cc}\hat{\alpha}=\frac{\hat{b}\ue619\hat{z}\ue89e\phantom{\rule{0.3em}{0.3ex}}}{\hat{z}\xb7\left(\hat{a}\ue619\hat{b}\right)}& \left(36\ue89e\text{-}\ue89e5\right)\\ \hat{\beta}=\frac{\hat{z}\ue619\hat{a}}{\hat{z}\xb7\left(\hat{a}\ue619\hat{b}\right)}& \left(36\ue89e\text{-}\ue89e6\right)\end{array}$
where â and {circumflex over (b)} are defined as
â=a_{x }{circumflex over (x)}+a_{y }ŷ (36-7)
{circumflex over (b)}=b_{x }{circumflex over (x)}+b_{y }ŷ (36-8)
Multiplying (36-5) and (36-6) on the second and third terms of Eqn. (34), the coordinate transformation is obtained
$\begin{array}{cc}r=\frac{{\mathrm{pb}}_{y}-{\mathrm{qb}}_{x}}{V}& \left(37\right)\\ s=\frac{-{\mathrm{pa}}_{y}+{\mathrm{qa}}_{x}}{V}& \left(38\right)\\ \mathrm{where}\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89eV={a}_{x}\ue89e{b}_{y}-{a}_{y}\ue89e{b}_{x}.& \left(39\right)\end{array}$
The coordinates of x-y system in Eqns. (32) and (33) are now transformed into those of an a-b system, as follows: Inserting Eqns. (36-3) and (36-4) into (37), (38), an explicit form is obtained:
$\begin{array}{cc}r=\frac{\alpha \ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\Omega}_{a}\ue89et\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e{b}_{y}\mp \alpha \ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{sin}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\Omega}_{a}\ue89et\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e{b}_{x}}{V}& \left(40\right)\\ s=\frac{-\alpha \ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\Omega}_{a}\ue89et\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e{a}_{y}\pm \alpha \ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\mathrm{sin}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\Omega}_{a}\ue89et\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e{b}_{x}}{V}& \left(41\right)\end{array}$
In summary, when the Ports P and Q are spaced apart with a general angle defined by Eqns (36-7) and (36-8) as shown in FIG. 3 or 5, a slowly rotating microwave field of rotation frequency of Ω_{a }can be excited by microwave field inputs from Ports P and Q represented by:
ζ_{Pa}=r cos(ωt+φ_{h}) (42-1)
ζ_{Qb}=s cos(ωt+φ_{h}) (42-2)
where r and s are defined in Eqns. (40) and (41), and the plus and minus sign of Eqn. (41) corresponds to anticlockwise and clockwise rotations, respectively. The forms of (42-1) and (42-2) are of the form of amplitude modulation with time varying functions of r and s.
Relating to Eqns. (22) and (28), we shall illustrate the case that Port Q is separated from port P by
$\frac{5\ue89e\pi}{4}$
to make a slow rotation of TE_{121 }as shown in FIG. 5:
$\begin{array}{cc}\hat{a}={a}_{x}\ue89e\hat{x}+{a}_{y}\ue89e\hat{y}=1\ue89e\hat{x}+0\ue89e\hat{\phantom{\rule{0.3em}{0.3ex}}\ue89ey}& \left(43\right)\\ \hat{b}={b}_{x}\ue89e\hat{x}+{b}_{y}\ue89e\hat{y}=\left(-\frac{1}{\sqrt{2}}\right)\ue89e\hat{x}+\left(-\frac{1}{\sqrt{2}}\right)\ue89e\hat{y}.& \left(44\right)\end{array}$
Substitution of Eqns. (43) and (44) into Eqns. (39)-(41) yields:
r=α cos Ω_{a}t−α sin Ω_{a}t (45)
s=−√{square root over (2)}α sin Ω_{a}t. (46)
This shows that, for the geometrical configuration of FIG. 5, when supplying microwave power in the form of Eqns. (45) and (46) from Ports P and Q respectively, the excited wave will be equal to that of Eqn. (33). This is verified by the fact that substitution of Eqns. (43)-(46) into Eqn. (34) yields Eqn. (36-2), which leads to Eqn. (32), eventually to Eqn. (33). For other configurations of Ports P and Q, the skilled worker can derive supplied powers of each port in the same manner as the foregoing.
Impedance Shifting by Irises in a Power-Supplying Waveguide:
Each of the two waveguides 360 of the embodiment of FIG. 2 or the embodiment of FIG. 3 has a radiation or coupling aperture 405 open through a respective one of the ports P and Q to the interior of the cylindrical cavity 120, as shown in FIG. 6. In the embodiment depicted in FIG. 6, the waveguide 360 is rectangular, having four conductive walls forming a rectangular cross-section, including a pair of side walls 410, 411, a floor 412 and a ceiling 413. An input opening 415 of the waveguide 360 is open for receiving microwaves. An opposite end 416 is covered by a wall 418. The coupling aperture 405 referred to above is formed in the wall 418 and is aligned with a corresponding one of the ports P and Q. Each port P and Q is an opening in the side wall of the cavity 120 and may match the dimensions of the coupling aperture 405.
The waveguide 360 may include one or more irises such as an iris 420. The iris 420 is formed as a rectangular window in a rectangular wall 422. Behavior of the waveguide 360 is determined by the dimensions of the rectangular input opening 415, a×b, the dimensions of the rectangular iris 420, c×d, the dimensions of the rectangular coupling aperture 405, e×f, the distance g between the iris 420 and the input end 415 and the distance h between the iris 420 and the coupling aperture 405. Other suitable shapes and dimensions can be chosen. To tune a chamber impedance, the coupling aperture size e×f is first adjusted. In one example, the best spectrum of a resonance “1” was obtained for e×f=60 mm×2 mm. For brevity of explanation, only the resonance “1” will be considered hereinafter.
Next, an arbitrary distance h of the iris 420 from the coupling aperture 405 is chosen. In FIG. 7, a capacitive iris is chosen, and the value of the dimension d is adjusted. As the size of d is changed, the impedances of three resonances move. Furthermore, its quality factor Q represented by the size of a resonant circle in a Smith chart also changes. FIG. 8 depicts an inductive iris, with which different impedance shifts may be obtained. A resonant iris shown in FIG. 9 can make a critical coupling where the chamber impedance of a target frequency is located on the center of a Smith chart. In this configuration, the stub tuners of FIGS. 2 and 3 can be removed, leading to a simplified and cost-effective chamber design. However, since the critical coupling has a high Q value, this setting will generally deteriorate repeatability of tuning.
As indicated in dashed line, a second iris plate 500 can be placed in the waveguide 360 to obtain a preferable chamber impedance. A third iris plate may be added as well.
Advantages:
A principal advantage of the embodiment of FIGS. 1-5 is that a rotating microwave for plasma processing can be produced for any suitable combination of user-selected mode indices m, n and l of modes TE_{mnl }and TM_{mnl }by suitable adjustment of spatial and temporal separation between microwave excitations at two different azimuthal locations. A principal advantage of the embodiment of FIGS. 6-9 is that the microwave chamber impedance can be adjusted without changing the chamber by introducing impedance shifting irises into the power-feeding waveguides coupled to the cavity.
While the foregoing is directed to embodiments of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.