Abstract:
The present invention generally relates to an evanescent microwave spectroscopy probe and methods for making and using the same. Some embodiments relate to a probe in electrical communication with sapphire tuning capacitors that are arranged in parallel. Some embodiments relate to using capacitors arranged in this manner to achieve higher Q values. Furthermore, probe can be used in microwave microscopy applications, and for imaging samples thereby.

Description:
RELATED APPLICATION DATA  
       [0001]     This application is a continuation-in-part of, and hereby claims priority to, U.S. patent application Ser. No. 11/255,497 filed on Oct. 20, 2005, pending; which claims priority to U.S. Provisional Patent Application No. 60/620,592 filed on Oct. 20, 2004; all of which are incorporated by reference in their entireties. 
     
    
     BACKGROUND OF THE INVENTION  
       [0002]     The present invention relates to probes for imaging evanescent microwave fields. In some embodiments the present invention relates to achieving high Q values by using sapphire capacitors that are arranged in parallel. Additionally, the present invention relates to methods of making and using such probes. Furthermore, some embodiments relate to using such probes for microwave microscopy, and/or for imaging samples thereby.  
         [0003]     Prior work in this area used a shunt series combination. Thus, the maximum Q was solely determined by the resistance of the series R-L-C probe equivalent circuit and tuning network. Additionally, prior methods used calculations based on capacitance arising from the gap between a spherical conducting tip and a perfectly conductive sample surface. As a result, such methods do not accurately predict how the probe reacts in the electric field between it and the sample.  
         [0004]     In contrast to prevailing methods, the method of the present invention is independent of the electrical properties of the material. Thus, unlike the prior art, the present invention applies equally well to dielectrics, conductors and superconductors. Furthermore, the method of the present invention, as set forth herein, enables the solution of the classical electrodynamic boundary value problem concerning a superconductor modeled as a dielectric having a complex permittivity with a large negative real part, which can be associated with the persistent current. Still further, in some embodiments the resistance is cut by up to about 50% in comparison to prior microwave probes, which results in higher Q values and correspondingly high sensitivity. Thus, the present invention represents a significant advance in the state of the art.  
       BRIEF SUMMARY OF THE INVENTION  
       [0005]     The present invention relates to near field microscopy and, more particularly to an evanescent microwave microscopy probe for use in near field microscopy and methodology for investigating the complex permittivity of a material through evanescent microwave technology. In one embodiment, the probe comprises a low loss, apertured, coaxial resonator that can be tuned over a large bandwidth by a parallel shunt sapphire tuning network. In one embodiment, the transmission line of the probe utilizes high grade paraffin, offering a low loss tangent and a very close dielectric match within the line. A chemically sharpened probe tip extends slightly past the end aperture of the probe and emits a purely evanescent field. The probe is extremely sensitive, achieving Q values in excess of 0.5×10 6  and a spatial resolution of 1.0×10 −6  meters.  
         [0006]     The physical construction of a probe according to the present invention results in a purely evanescent field emanating from its tip. As a result, when a probe of the present invention is used in quantitative microscopy, it is not necessary to use additional hardware and/or a methodology to separate a propagative component from the field. Probes of the present invention also provide an extremely low loss impedance match to standardized equipment. The low loss coaxial resonator of the present invention theoretically has an infinite bandwidth but in practice its bandwidth is governed by its physical length and the source bandwidth. In the present invention the evanescent mode bandwidth is controlled by the aperture diameter, which is quite large compared with state of the art designs.  
         [0007]     The probe of the present invention also utilizes a shunt capacitive tuning network characterized by a low equivalent circuit resistance. As a result, the probe of the present invention provides for large resonant frequency selection range and extremely high Q values.  
         [0008]     In some embodiments, the present invention relates to an evanescent microwave microscopy probe, comprising: a center conductor having a first end and a second end, wherein the center conductor comprises a waveguide for microwave radiation; a probe tip affixed to the first end of the center conductor, wherein the tip is capable of acquiring a near-field microwave signal from a sample; an outer shield surrounding the center conductor, wherein the center conductor and outer shield are in a generally coaxial relationship, wherein the outer shield has a first end and a second end corresponding to the first and second ends of the center conductor, and wherein the center conductor and outer shield are not in direct contact and thereby form a gap; an insulating material occupying at least a portion of the gap between the center conductor and the outer shield; an aperture located near the tip, wherein the aperture comprises a plate having an inside face and an outside face, wherein the aperture is oriented generally perpendicular to the center conductor, and wherein the aperture comprises a hole that allows the tip to be in microwave communication with a sample; and a tuning network in electronic communication with the second end of the center conductor and with the outer shield, wherein the tuning network comprises a pair of capacitors in a parallel electronic relationship.  
         [0009]     The present invention also relates to a process for making a microwave probe comprising: providing a center conductor having a first end and a second end, wherein the center conductor comprises a waveguide for microwave radiation; affixing a probe tip to the first end of the center conductor, wherein the tip is capable of acquiring a near-field microwave signal from a sample; surrounding the center conductor with an outer shield, wherein the center conductor and outer shield are in a generally coaxial relationship, wherein the outer shield has a first end and a second end corresponding to the first and second ends of the center conductor, and wherein the center conductor and outer shield are not in direct contact and thereby form a gap; occupying at least a portion of the gap between the center conductor and the outer shield with an insulating material; providing an aperture located near the tip, wherein the aperture comprises a plate having an inside face and an outside face, wherein the aperture is oriented generally perpendicular to the center conductor, and wherein the aperture comprises a hole that allows the tip to be in microwave communication with a sample; and providing a tuning network in electronic communication with the second end of the center conductor and with the outer shield, wherein the tuning network comprises a pair of capacitors in a parallel electronic relationship.  
         [0010]     Further, the present invention relates to a method for detecting a sample using an electromagnetic microwave field comprising: providing the probe of claim  1 ; obtaining a resonant frequency reference reading from the probe, wherein the probe is substantially decoupled from a sample; placing the probe of claim  1  in electromagnetic microwave communication with the sample; obtaining an resonant frequency reading from the probe; calculating a resonant frequency change relative to the reference reading; and relating the resonant frequency change to one or more properties of the sample.  
         [0011]     Still further, the present invention relates to a method for imaging a sample using an electromagnetic microwave field comprising: providing the probe of claim  1 ; providing an X-Y sample stage, and a sample disposed thereon; obtaining a resonant frequency reference reading from the probe, wherein the probe is substantially decoupled from the sample; placing the probe of claim  1  in electromagnetic microwave communication with the sample at a first position; obtaining a resonant frequency reading from the probe at the first position; moving the probe to a next position; obtaining a resonant frequency reading from the probe at the next position; repeating the preceding two steps as needed; calculating a resonant frequency change at each position relative to the reference reading; and plotting an image of the sample as a function of position and frequency change or a property related to frequency change. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0012]      FIG. 1  is a schematic representation of a cross-section of a probe in accordance with the present invention;  
         [0013]      FIG. 2  is a block diagram of a microscope in accordance with the present invention;  
         [0014]      FIG. 3  is a diagram of the probe and coupling network;  
         [0015]      FIG. 4  is a diagram showing a method of images;  
         [0016]      FIG. 5  is a scanning electron micrograph of a superconducting film having two distinct regions;  
         [0017]      FIG. 6  is a plot of susceptibility loss versus temperature for a superconducting film;  
         [0018]      FIG. 7  is a pair of plots of resonant frequency versus distance between the probe tip and the sample, wherein the data is collected at 79.4 K and 298 K;  
         [0019]      FIG. 8  is a plot showing the change in Q for the superconducting film at 79.4 K;  
         [0020]      FIG. 9  is a photograph of an embodiment of a microwave microscopy apparatus in accordance with the present invention;  
         [0021]      FIG. 10  is a photograph of Ti—Au lines etched on sapphire at 20× magnification;  
         [0022]      FIG. 11  is a plot of a change in Q;  
         [0023]      FIG. 12  is a plot of a change in reflection coefficient images;  
         [0024]      FIG. 13  is a circuit diagram representing a probe connected to a superconductor;  
         [0025]      FIG. 14  is a plot showing a change in Q for a superconducting film in junction area of 6° bi-crystal;  
         [0026]      FIG. 15  is a plot showing the tuned resonance with a probe tip one micron above a SrTiO 3  crystal sample at 300 K;  
         [0027]      FIG. 16  is a plot showing the frequency-shifted resonance with a probe tip about 1 micron from a SrTiO 3  crystal sample at 302 K; and  
         [0028]      FIG. 17  is a drawing of a chamfered aperture having a ceramic coating in place of a poly(tetrafluoroethylene) plug. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0029]     The present invention generally relates to a microwave probe for microwave microscopy and to a method of using the same for generating high quality microwave data. More particularly, the apparatus and method of the present invention can be used to take high-precision, low-noise, measurements of material parameters such as permittivity, permeability, and conductivity.  
         [0030]     The probe can be used for the characterization of local electromagnetic properties of materials. The resonator-intrinsic, spatial resolution is experimentally demonstrated herein. A first-order estimation of the sensitivity related to the probe tip-sample interaction for conductors, dielectrics, and superconductors is provided. An estimation of the sensitivity inherent to the resonant probe is set forth. The probe is sensitive in a range of theoretically estimated values, and has micrometer-scale resolution.  
         [0031]     In the field of evanescent microwave microscopy, the tip of the probe operates in close proximity to the sample, where the tip radius and effective field distribution range are much smaller than the resonator excitation wavelength. The propagating field exciting resonance in the probe can be ignored and the probe tip-sample interaction can be treated as quasi-static. This can be used for localized measurements and images with resolved features governed essentially by the characteristic size of the tip. The field distribution from the probe tip extends outward a short distance, and as the sample enters the near field of the tip, it interacts with the evanescent field, thus perturbing the probe&#39;s resonance. This perturbation is linked to the resonant structure of the probe through the air gap coupling capacitance C C  between the tip and the material. This results in the loading of the resonant probe and alters the resonant frequency f r , quality factor Q, and reflection coefficient S 11  of the resonator.  
         [0032]     If the air gap distance from tip to sample is held constant, the f r , Q, and S 11  variations related to the microwave properties of the sample can be mapped as the probe tip is scanned over the sample. The microwave properties of a material are functions of permittivity ∈, permeability μ, and conductivity σ.  
         [0033]     Referring to  FIG. 1 , the microwave probe  10  of the present invention can be constructed from a 0.085″ semi-ridged coaxial transmission line. The probe  10  is based on an end-wall aperture coaxial transmission line, where the resonator behaves as a series resonant circuit for odd multiples of λ/4.  
         [0034]     In constructing probe  10 , the center conductor is removed along with the poly(tetrafluoroethylene) insulator and replaced with, for example, high purity paraffin  14 . However, the invention is not restricted to paraffin and alternative materials can be used. For example, alternative materials within the scope of the present invention include, without limitation, magnesium oxide, titanium oxide, boron nitride, aluminas. Other waveguide materials include copper, aluminum, brass, alum, and any combination thereof. Still other wave guide materials include polytetrafluoroethylene (PTFE), PTFE/glass fabrics such as Taconic RF-35, and various organic polymeric materials and composites.  
         [0035]     Fashioning probe  10  according to the foregoing paragraph results in coaxial wave guide probe  10  rather than an open cavity. A copper aperture, having a thickness of about 0.010″, is soldered inside outer shield  15 , creating end-wall aperture  12 . Chemically sharpened probe tip  17  is mounted on center conductor  16  and electroplated with silver. The transmission line resonator is then reconstructed by casting the sharpened, plated, center conductor  16  inside the outer shield  15  with high purity paraffin  14 . A short section of the original poly(tetrafluoroethylene) shielding replaces paraffin  14  at the sharpened end of the coax, and is located directly above end-wall aperture  12 . Poly(tetrafluoroethylene) plug  18  is used to maintain tip-aperture alignment. Sharpened probe tip  17  of center conductor  16  extends beyond shielded end-wall aperture  12  of resonator by approximately 0.001″ or less. The purely evanescent probing field radiates from sharpened probe tip  17 . In this manner, as the radius of center conductor  16  decreases, the spatial resolution of the probe increases due to localization of the interaction between probe tip  17  and sample  20 .  
         [0036]     In an alternative embodiment, poly(tetrafluoroethylene) plug  18 , which is disposed at the aperture end, can be replaced with a ceramic, for instance a ceramic coating on the backside of the aperture. Such a coating is applied before aperture  12  is soldered to outer shield  15 . In one embodiment this is done using a high temperature strain gauge ceramic adhesive, known by the trade-name Ceramabond-671. In other embodiments the coating can be formed through pulsed laser deposition of any of a wide variety of suitable ceramics including, without limitation, cerium oxide. In one embodiment, the coated side of the aperture is optionally chamfered at about 60° prior to coating (see  FIG. 17 ). Alternatively, the chamfer can be between about 45° and about 75°. This practice results in an increase in Q, and reduced reflection.  
         [0037]     Referring to  FIG. 2 , the microwave excitation frequency of resonant probe  10  can be varied over a bandwidth from about 1 to 40 GHz in network analyzer  40 , and is tuned by external capacitors  30 . As further illustrated in  FIG. 3 , microscope probe  10  can be coupled to network analyzer  40  through tuning network capacitors C 1    31  and C 2    32 , which are connected to center conductor  16  and to outer shield  15 .  
         [0038]     A block diagram of the microwave microscopy system is shown in  FIG. 2 . The changes in the probe&#39;s resonant frequency, quality factor (Q), and reflection coefficient are tracked by a Hewlett-Packard 8722ES network analyzer  40  through S 11  port measurements, as probe  10  moves above sample surface  20 . The microwave excitation frequency of resonant probe  10  can be varied within the bandwidth of network analyzer  40  and tuned to critical coupling with tuning assembly  30 . Tuning assembly  30  comprises two variable 2.5 to 8 pF capacitors  31 ,  32 . The tuning network has one capacitor C 1    31  connected in-line with center conductor  16 , and the other capacitor C 2    32  is connected from center conductor  16  to ground.  
         [0039]     The X-Y axis stage  70  is driven by Coherent® optical encoded DC linear actuators. Probe  10  is frame-mounted to a Z-axis linear actuator assembly and the height at which probe  10  is above sample surface  20  can be precisely set. The X-Y stage actuators, network analyzer  40 , and data acquisition and collection are controlled by computer  50 . The program that interfaces to the X-Y stage actuators, serial port communications, 8722ES GPIB interface, and data acquisition is written in National Instruments Labview® software. The complete evanescent microwave scanning system is mounted on a vibration-dampening table (see  FIG. 9 ).  
         [0040]     According to one embodiment of the present invention, external tuning capacitor assembly  30  comprises two thermally compensated sapphire capacitors in a shunt configuration. If a shunt is placed near the end of the resonator then the Q of the resonator will theoretically approach infinity. Sapphire capacitors are advantageous because they exhibit frequency invariance up to approximately 10 GHz. In one embodiment, the capacitors  31 ,  32  are variable from, for example, about 4.5 to 8.0 Picofarads. The position of capacitors  31 ,  32  in tuning assembly  30  is optimized to reduce interaction. Shielding techniques can also be employed to limit external interaction and leakage.  
         [0041]     As is noted above, the present invention also relates to methodology for investigating the complex permittivity of a material through evanescent microwave technology. More particularly, the methodology taught herein is a scheme for investigating the complex permittivity of a material, independent of its other electrical properties, through evanescent microwave spectroscopy.  
         [0042]     The extraction of quantitative data through evanescent microwave microscopy requires a detailed configuration of the field outside the probe-tip region. The solution of this field relates the perturbed signal to the distance between the probe-tip and sample, and physical material properties. In accordance with the present invention, the mode of the field generated at the tip is evanescent. A mixed mode field including evanescent and propagative components impedes quantitative measurements. The propagative wave&#39;s contribution to the analytical signal depends on the electrical properties of the sample, and limits resolution.  
         [0043]     In analyzing conductors quantitatively the probe tip can be modeled as a conducting sphere and the sample as an ideal conductor. The separation between the tip and sample can be modeled as a capacitor with capacitance C c , resulting in a resonant frequency shift that is proportional to the variation in C c . When a conducting material is placed near the tip electrical interaction therewith causes charge and field redistribution. The method of images can be applied to model this redistribution of the field using a series iteration of two image charges. This variation in tip-sample capacitance results in a detectable shift in resonant frequency of probe  10 .  
         [0044]     As noted above, the method of images can be applied to quantitatively analyze dielectric materials. Additionally, the probe tip is modeled as a charged conducting sphere with potential V 0 . When the tip is placed close to a dielectric material the dielectric is polarized by the tip&#39;s electric field. This dielectric reaction to the tip causes a redistribution of charge on the tip in order to maintain the equipotential surface of the sphere, and also results in a shift in the probe&#39;s frequency. The method of images can be applied to model the field redistribution using a series of three image charges in an iterative process to meet boundary conditions at the tip of probe  10  and dielectric sample surface  20 .  
         [0045]     In this unified approach, perturbation theory for microwave resonators is applied to the field distribution outside the tip. The expression for the resonant frequency shift due to the presence of a sample material can be written as  
                   Δ   ⁢           ⁢   f       f   0       =       -         ∫   V     ⁢       [         (     Δ   ⁢           ⁢   ɛ     )     ⁢     (       E   _     ·       E   _     0       )       +       (     Δ   ⁢           ⁢   μ     )     ⁢     (       H   _     ·       H   _     0       )         ]     ⁢     ⅆ   V             ∫   V     ⁢       (         ɛ   0     ⁢       E   _     0   2       +       μ   0     ⁢       H   _     0   2         )     ⁢     ⅆ   V             =       f   -     f   0         f   0           ,           (   1   )             
 
 where  E  and  H  are the perturbed fields, V is the volume of a region outside the resonator tip, f is the resonant frequency and f 0  is the reference frequency. The unperturbed field can be obtained by  
                     E   0     ⁡     (     r   ,   z     )       =       q     4   ⁢     πɛ   0         ⁢       [       r   ⁢           ⁢     r   ^       +       (     z   +       a   1   ′     ⁢     r   0         )     ⁢     z   ^         ]         [       r   2     +       (     z   +       a   1   ′     ⁢     r   0         )     2       ]       3   /   2             ,           ⁢              H   _     0          =           ɛ   0       μ   0         ⁢            E   _     0                ⁢     
     ⁢   where           (   2   )                   a   1   ′     =       r   0     +   g       ,           (   3   )             
 
 and where r 0  is the radius of the spherical tip and g is the gap between the tip and surface of sample  20 . The potential V 0  on the spherical tip is given by  
               V   0     =       q     4   ⁢     πɛ   0     ⁢     r   0         .             (   4   )             
 
         [0046]     By using the method of images (see  FIG. 4 ), the perturbed electric field in the region between the tip and sample, and within the sample volume (assuming r 0  is much smaller than the sample thickness) can be modeled as  
                     E   _     1     ⁡     (     r   ,   z     )       =       q     4   ⁢     πɛ   0         ⁢       ∑     n   =   1     ∞     ⁢       q   n     ⁢     {         [       r   ⁢           ⁢     r   ^       +       (     z   +       a   n   ′     ⁢     r   0         )     ⁢     z   ^         ]         [       r   2     +       (     z   +       a   n   ′     ⁢     r   0         )     2       ]       3   /   2         -     b   ⁢       [       r   ⁢           ⁢     r   ^       +       (     z   -       a   n   ′     ⁢     r   0         )     ⁢     z   ^         ]         [       r   2     +       (     z   -       a   n   ′     ⁢     r   0         )     2       ]       3   /   2             }             ,              H   _     1          =           ɛ   0       μ   0         ⁢            E   _     1              ,           (   5   )                       E   _     2     ⁡     (     r   ,   z     )       =       q     4   ⁢     π   ⁡     (     ɛ   +     ɛ   0       )           ⁢       ∑     n   =   1     ∞     ⁢       q   n     ⁢       [       r   ⁢           ⁢     r   ^       +       (     z   +       a   n   ′     ⁢     r   0         )     ⁢     z   ^         ]         [       r   2     +       (     z   +       a   n   ′     ⁢     r   0         )     2       ]       3   /   2                 ,           ⁢              H   _     2          =         ɛ   μ       ⁢            E   _     2              ,           (   6   )             
 
 where μ is real and  
                 a   n   ′     =       a   1   ′     -     1       a   1   ′     +     a     n   -   1     ′             ,           ⁢       q   n     =       t   n     ⁢   q       ,           ⁢       t   n     =       bt     n   -   1           a   1   ′     +     a     n   -   1     ′           ,     
     ⁢       t   1     =   1     ,           ⁢     b   =       ɛ   -     ɛ   0         ɛ   +     ɛ   0           ,           ⁢     ɛ   =       ɛ   ′     +     ⅈ   ⁢           ⁢       ɛ   ″     .                   (   7   )             
 
         [0047]     Importantly, for a tip in free space ∈=∈ 0  and μ=μ 0  at the location r=0 and z=−g−r 0 ,  E   0 =  E   1 =  E   2  and  H   0 =  H   1 =  H   2 , confirming the asymptotic behavior in equations (2), (5), and (6). By integrating the unperturbed electric field in equation (2) and the perturbed electric fields in equations (5) and (6) over a region V outside the spherical tip the frequency shift of equation (1) becomes  
                   (       Δ   ⁢           ⁢   f       f   0       )     TOTAL     =         (       Δ   ⁢           ⁢   f       f   0       )     1     +       (       Δ   ⁢           ⁢   f       f   0       )     2     -     A   ⁢       ∑     n   =   1     ∞     ⁢       t   n     ⁢     {     1   -       1   2     ⁢     (     1   -   b     )     ⁢     1       a   1   ′     +     a     n   -   1     ′             }           -       A   ⁡     (     Δμ   Δɛ     )       ⁢       ɛ   μ       ⁢         ɛ   0       μ   0         ⁢       ∑     n   =   1     ∞     ⁢       t   n     ⁢     b       a   1   ′     +     a     n   -   1     ′                   ,           ⁢     (     A   =     A   ′       )     ,     
     ⁢   where           (   8   )                       (       Δ   ⁢           ⁢   f       f   0       )     1     =       -     A   ′       ⁢       ∑     n   =   1     ∞     ⁢       t   n     ⁢     {     1   -       1   2     ⁢     (     1   -   b     )     ⁢     1       a   1   ′     +     a     n   -   1     ′             }             ,           ⁢     Reg   .           ⁢   A     ,           ⁢     Δμ   =   0       ⁢     
     ⁢   and           (   9   )                     (       Δ   ⁢           ⁢   f       f   0       )     2     =       -     A   ⁡     (     1   +       Δμ   Δɛ     ⁢       ɛ   μ       ⁢         ɛ   0       μ   0             )         ⁢       ∑     n   =   1     ∞     ⁢       t   n     ⁢     b       a   1   ′     +     a     n   -   1     ′                 ,           ⁢     Reg   .           ⁢   B   .             (   10   )             
 
         [0048]     Parameters A and A′ are constants determined by the geometry of the tip-resonator assembly. Taking into account the real part of equation (8), the analytical expression, fits with the experimental data below.  
         [0049]     As noted above, prior methods used calculations based on capacitance arising from the gap between a spherical conducting tip and a perfectly conductive sample surface. As a result, such methods do not accurately predict how the probe reacts in the electric field between it and the sample. The method of the present invention overcomes this deficiency. Moreover, the results of the prior art can be reproduced by the present method if the following additional restrictions are imposed on the reaction of the resonator probe to electric fields outside the tip. Namely, the coefficients in equations (9) and (10) must be equal (A′=A). This assumption provides a smooth transition between insulators and ideal conductors by assuming b=1 in equation (8).  
         [0050]     In one embodiment the method of the present invention is used to measure the dielectric properties of the superconductor YBa 2 Cu 3 O 7-δ . A superconductor can be treated as a dielectric material with a negative dielectric constant rather than a low loss conductor. In this embodiment probe  10  comprises a tuned, end-wall apertured coaxial transmission line. Resonator probe  10  is coupled to network analyzer  40  through tuning network  30  and coupled to sample  20  (see  FIG. 2 ). When probe tip  17  is in close proximity to sample  20 , the resonator&#39;s frequency f shifts. In measuring the frequency shift, the resonant frequency reference is set with probe tip  17  at a fixed distance above sample  20 . The distance between probe tip  17  and sample  20  is sufficient to assure that the evanescent field emanating from tip  17  will not interact with sample  20 . The field dispersion from the probe tip extends outward a short distance with the amplitude of the evanescent field decaying exponentially. As sample  20  enters the near field of probe tip  17  it interacts with the evanescent field, thereby perturbing it. This results in loading the probe  10 . Accordingly, sample  20  is considered part of the resonant circuit and results in losses to the system, which decreases the probe&#39;s  10  resonant frequency. The measured frequency shift, as it relates to tip-sample separation g, generates a transfer function relating Δf to Δg. The transfer function is best fit with an electrostatic field model using the method of images to extract complex permittivity values.  
         [0051]     In one variation of the foregoing embodiment, the evanescent microwave microscopy system is adapted for making cryogenic measurements. A miniature single-stage Joule-Thompson cryogenic system is fixed to X-Y stage  70 . Microwave probe  10  is fitted through a bellows, which provides a vacuum seal and allows the probe to move freely over sample  20 , which is mounted on a cryogenic finger directly below probe  10 .  
         [0052]     In this embodiment, an YBa 2 Cu 3 O 7-δ  superconducting thin film is fabricated by pulsed laser deposition. This deposition method results in two distinct regions,  1  and  2 , forming on a 0.5 mm thick LaAlO 3  substrate (see  FIG. 5 ). The superconductive transition temperatures for region  1  and  2  of the film are T c =92 K and 90 K respectively, which are measured by plotting susceptibility loss versus temperature under different amplitudes of alternating magnetic field at the frequency of 2 MHz, as shown in  FIG. 6 . The measured frequency shift data is collected for both regions at 79.4 K and 298 K as shown in  FIG. 7 . Fitting parameters from equation (8) to the experimental data are consolidated in Table I.  
                                                                   TABLE I                           SIMULATION FIT PARAMETERS FOR YBa 2 Cu 3 O 7     τ     δ         SUPERCONDUCTING THIN FILM AT 79.4 K AND 298 K.                A   ε′/ε 0         r 0     μ/μ 0         REGIONS   (10 −4 )   (10 8 )   ε″/ε 0     (10 −6 m)   (10 −4 )                    REGION 1 at 79.4 K   2.09   −9.2   −0.1   3.35   1       REGION 2 at 79.4 K   2.08   −9   −0.1   3.35   1       TRANSITION REGION   2.08   −9.1   −0.1   3.35   1       at 79.4 K       REGION 1 at 298 K   1.45   1   6.6   8   1       REGION 2 at 298 K   1.45   1   6.85   8   1                  
 
         [0053]     Above the transition temperature (T c ), the superconductor behaves like a metallic conductor. Thus, the sign and magnitude of the real and imaginary permittivity values change (Table I).  FIG. 7  shows the curves from both regions below T c  and illustrates that there is a distinct measurable difference between these regions. The transition section connecting regions  1  and  2  with the associated frequency shift fit parameters generated at 79.4 K falls in between fit curves for regions  1  and  2 . The model fit parameters for this transition segment are A=2.08×10 −4 , which is the resonator scaling factor, the real component of permittivity ∈=−9.13×10 8   ∈     0   , the imaginary component of permittivity ∈″=−0.1 ∈     0   , and the effective tip radius r 0 =3.35 μm.  FIG. 8  shows a change in Q scan performed at 79.4 K over both regions, and indicates the average dynamic range of Q in this scan between the two areas is approximately 8000. The higher Q level is associated with the area of T c =92 K, and the lower Q level corresponds to region of T c =90 K.  
         [0054]     The resolution of the probe is verified using a sapphire polycrystalline substrate with titanium-gold etched lines of widths ranging from about 10 μm to 1 μm (see  FIG. 10 ). Titanium is used for adhesion of the gold to the substrate, and is approximately 100 nm thick, while the thickness of the gold deposition is approximately 1 μm. The resonant frequency of the probe is tuned to 2.67 GHz. The etched lines of the sample are scanned with the probe resulting in a change in frequency, Q, and magnitude of reflection plots.  
         [0055]     The smallest physically resolvable feature for an evanescent probe is governed by the size of the tip radius, along with the height at which the tip is positioned above the feature. For example, to resolve a 5 μm physical feature, the probe tip radius r 0  must be approximately 5 μm or less from tip to sample.  
         [0056]     The change in Q and change in magnitude of reflection coefficient images are illustrated in  FIGS. 11 and 12 , respectively. The data for these plots are taken from a 20 μm×18 μm scan area around a 1 μm wide etched line. The measured tip radius of the probe used is 1.2 μm with a stand off height (g) of 2 μm and a 1 μm data acquisition step. The location of the etched line is indicated on each plot by arrows with corresponding measurements in micrometers. In this embodiment, the one micrometer line was distinguishable in both plots, which gives the probe about 1 μm topographical resolution or better. The Q values attainable with this tunable probe range from 1.5×10 4  to well over 10 5 , and even over 10 6 . According to this embodiment, the dynamic range of the change in Q is approximately 5×10 5  , as shown in  FIG. 11 .  
         [0057]     The Johnson noise-limited sensitivity is analyzed in the present invention by setting the signal power equal to the noise power resulting in [(δ∈/∈)]=2.45×10 −5 . As those of ordinary skill in the art are aware, Johnson noise results from random thermal movements of charge carriers, and is often referred to alternatively as thermal noise.  
         [0058]     The sensitivity of the evanescent microwave probe described herein can be separated into two categories. The first, S r , is inherent to the resonator itself and directly proportional to it&#39;s quiescent operating value Q. The other, S f , is external to the resonator and solely determined by tip-sample interactions. A noise threshold has to be considered in an evanescent microwave system, which also affects sensitivity.  
         [0059]     The minimum detectable signal in an evanescent microwave microscopy system should be greater than the noise created by the probe, tuning network, and coupling to the sample. The noise is generated by a resistance at an absolute temperature of T by the random motion of electrons proportional to the temperature T within the resistor. This generates random voltage fluctuations at the resistor terminal, which has a zero average value, but a non-zero rms value given by Planck&#39;s&#39;black body radiation law. These voltage fluctuations can be calculated by the Raleigh-Jeans approximation as
 
 V   n(rms) =√{square root over (4 kTBR )},  (11)
 
 where k=1.38×10 −23  J/K is Boltzmann&#39;s constant, T is the temperature in Kelvin, B is the bandwidth of the system in Hertz, and R is the resistance in ohms. The resistance that results at critical coupling is the resistance R that produces noise in the system. Therefore, the signal level should be above this noise level in order to be detectable. 
 
         [0060]     The sensitivity approximation internal to the resonator S r  can be determined theoretically and experimentally. The theoretical value is analytically approximated by considering the lumped series equivalent circuit of the resonator, which has an inherent resonant frequency ω 0  and Q associated with the lumped parameters R 0 , L 0 , and C 0 . This configuration and associated parameters can be viewed as if the probe tip is beyond the decay length of the evanescent field from a material, or in free space. If the probe tip is brought into close proximity and electrically couples to the sample, the resonant frequency ω 0  and Q are perturbed to a new value ω′ 0  and Q′, respectively, and are associated with new perturbed parameters R′ 0 , L′ 0 , and C′ 0 . The total impedance looking into the terminals of the perturbed resonator coupled to a sample can be written as  
               Z   TOTAL     =         R   0   ′     ⁡     [     1   +     j   ⁢           ⁢     Q   ⁡     (       ω     ω   0   ′       -       ω   0   ′     ω       )           ]       .             (   12   )             
 
         [0061]     The magnitude of the reflection coefficient S 11  is related to Z TOTAL  by  
                 S   11     =         Z   TOTAL     -     Z   0           Z   TOTAL     +     Z   0           ,           (   13   )             
 
 where Z 0  is the characteristic impedance of the resonant structure. If one assumes critical coupling, where the resonator is matched to the characteristic impedance of the feed transmission line at resonant frequency, then R′ 0 ≈Z 0  at ω≈ω′ 0  and S r  is defined in as  
                 S   r     =         ⅆ     S   11         ⅆ   ω       ≈         Q   ′       ω   0   ′       ⁢     (     1   -     Δω     ω   0   ′         )           ,           (   14   )             
 
 where Δω=ω−ω′ 0 . 
 
         [0062]     The external sensitivity determined by tip-sample interaction of probe  10  is based on a λ/4 section of transmission line, with the lumped parameter series equivalent circuit coupled to an equivalent circuit model of a superconductor shown in  FIG. 7 . The series lumped parameter circuit for the resonator consists of R 0 , L 0 , and C 0  and the probe tip coupling to the superconductor is represented by C C . The equivalent circuit model of the superconductor is comprised of R S , L S , C S , and L C , where the series combination of R S  and L S  represents the normal conduction. The element L C  signifies the kinetic inductance of the Cooper-pair flow and C S  is related to displacement current. The superconductor equivalent circuit contains the necessary circuit elements in the appropriate configuration to represent not only a superconductor, but a metallic conductor and a dielectric.  
         [0063]     The equivalent circuit model for the probe coupled to a superconductor is illustrated in  FIG. 7 , where the equivalent circuit model for the superconductor is derived from the two-fluid model. The lumped circuit representation of the superconductor comprises capacitance C S , the inductance for normal carrier flow L S , and resistivity ρ=1/σ 1 , shunted by kinetic inductance L C =1/ωσ 2 . The parameters C S  and L S  are considered to have minimal effects when the superconductor is subjected to low frequencies and is neglected in this analysis. The conductivity ratio y=σ 1 /σ 2  is correlated to the impedance ratio y=ωL C /ρ and in the limit of large y (y&gt;&gt;1), σ 2 =0 and L C &gt;&gt;1. The opposite extreme, y&lt;&lt;1 results in L C  approaching 0, while σ 2  advances toward infinity. The superconductive samples for this study were subjected to a frequency of approximately 1 GHz and are of an inductive nature. The superconductor with an inductive nature has L C &lt;&lt;R S .  
         [0064]     The impedance Z 1  is the parallel combination of R S  and L C  and is represented as  
               Z   1     =         jω   ⁢           ⁢     L   C     ⁢     R   S           R   S     +     jω   ⁢           ⁢     L   C           .             (   15   )             
 
         [0065]     The impedance Z 2  is the series combination of C C  and Z 1,  which results in  
               Z   2     =         1     jω   ⁢           ⁢     C   C         +       jω   ⁢           ⁢     L   C     ⁢     R   S           R   S     +     jω   ⁢           ⁢     L   C             =           R   S     +     jω   ⁢           ⁢     L   C       +     jω   ⁢           ⁢       C   C     ⁡     (     jω   ⁢           ⁢     L   C     ⁢     R   S       )             jω   ⁢           ⁢       C   C     ⁡     (       R   S     +     jω   ⁢           ⁢     L   C         )           .               (   16   )             
 
         [0066]     The impedance Z 3  is the parallel combination of Z 2  and C 0  given by  
                       1             ⁢     Z             ⁢   3           =       1             ⁢     Z             ⁢   2           +     jω   ⁢           ⁢     C             ⁢   0                       =         jω   ⁢           ⁢       C   C     ⁡     (       R   S     +     jω   ⁢           ⁢     L   C         )             R   S     +     jω   ⁢           ⁢     L   C       +     jω   ⁢           ⁢       C   C     ⁡     (     jω   ⁢           ⁢     L   C     ⁢     R   S       )             +     jω   ⁢           ⁢     C   0                 ⁢     
     ⁢             Z   3     =         R   S     -       ω   2     ⁢     L   C     ⁢     C   C     ⁢     R   S       +     jω   ⁢           ⁢     L   C             jω   ⁢           ⁢       C   C     ⁡     (       R   S     +     jω   ⁢           ⁢     L   C         )         +       C   0     ⁡     (       R   S     -       ω   2     ⁢     L   C     ⁢     C   C     ⁢     R   S       +     jω   ⁢           ⁢     L   C         )                       =     -       j   ω     ⁡     [         R   S     -       ω   2     ⁢     L   C     ⁢     C   C     ⁢     R   S       +     jω   ⁢           ⁢     L   C             jω   ⁢           ⁢       C   C     ⁡     (       R   S     +     jω   ⁢           ⁢     L   C         )         +       C   0     ⁡     (       R   S     -       ω   2     ⁢     L   C     ⁢     C   C     ⁢     R   S       +     jω   ⁢           ⁢     L   C         )           ]                     =       -     j   ω       ⁢       Z   3   ′     .                       (   17   )             
 
         [0067]     The total impedance Z TOTAL  looking into the terminals of the probe coupled to a superconductor sample is  
         Z   total     =       R   0     +     jω   ⁢           ⁢     L   0       -       j   ω     ⁢       Z   3   ′     .             
 
         [0068]     The complex impedance Z 3  can be represented as  
         Z   3     =         1   jω     ⁡     [     Re   ⁡     (     Z   3   ′     )       ]       =     -         j   ω     ⁡     [     Re   ⁡     (     Z   3   ′     )       ]       .             
 
         [0069]     At resonance, the inductive and capacitive reactances cancel; therefore,  
                   jω   ⁢           ⁢     L   0       -       j   ω     ⁡     [     Re   ⁡     (     Z   3   ′     )       ]         =   0     ,           ⁢         ω   2     ⁢     L   0       =       Re   ⁡     (     Z   3   ′     )       .               (   18   )             
 
         [0070]     This allows one to solve for perturbed frequency ω in terms of the perturbed lumped circuit parameters in an iterative process, where one will be taking a first-order approximation. The combination of equations (7) and (8) results in  
                       ω   2     ⁢     L   0       =           R   S   2     ⁡     (     1   -       ω   2     ⁢     L   C     ⁢     C   C         )       ⁢     (       C   C     +     C   0     -       ω   2     ⁢     C   0     ⁢     L   C     ⁢     C   C         )             (       C   C     -     C   0     -       ω   2     ⁢     C   0     ⁢     L   C     ⁢     C   C         )     2     +       ω   2     ⁢         L   C   2     ⁡     (       C   C     +     C   0       )       2                       =       1     (       C   C     +     C   0       )       ⁢       (     1   -       ω   2     ⁢     L   C     ⁢     C   C       -       ω   2     ⁢     L   C     ⁢         C   0     ⁢     C   C           C   C     +     C   0             )       (     1   -     2   ⁢     ω   2     ⁢     L   C     ⁢         C   0     ⁢     C   C           C   C     +     C   0             )                     =       1     (       C   C     +     C   0       )       ⁢     1           (     1   -     2   ⁢     ω             ⁢   2       ⁢     L             ⁢   C       ⁢               ⁢       C             ⁢   0       ⁢           ⁢     C             ⁢   C                     ⁢       C             ⁢   C       ⁢           +           ⁢     C             ⁢   0                 )               [     1   +       ω   2     ⁢     L   C     ⁢       C   C     ⁡     (     1   +       C   0         C   C     +     C   0           )           ]                           =       1     (       C   C     +     C   0       )       ⁢       1     1   +       ω   2     ⁢     L   C     ⁢       C   C   2         C   C     +     C   0               .                     (   19   )             
 
         [0071]     Therefore, for the first iteration, one has the following equation  
               ω   0   ′2     =       1       L   0     ⁡     (       C   C     +     C   0       )         ⁢       1     [     1   +         L   C       L   0       ⁢       (       C   C         C   C     +     C   0         )     2         ]       .               (   20   )             
 
         [0072]     Solving for ω′ 0  in equation (20) results in  
                 ω   0   ′     =       ω   0     ⁢     1       1   +       C   C       C   0             ⁢     1         1   +       L   C       L   0           ⁢       (       C   C         C   C     +     C   0         )     2             ,           (   21   )             
 
 where  
           L   C       L   0       ⪡   1.       
 
         [0073]     The Taylor expansion of equation (21) gives  
               ω   0   ′     =           ω   0     ⁡     (     1   -       1   2     ⁢       C   C       C   0           )       ⁡     [     1   -       1   2     ⁢       L   C       L   0       ⁢       C   C   2         (       C   0     +     C   C       )     2           ]       .             (   22   )             
 
         [0074]     The sensitivity S f  for a superconductor is defined as  
                 S   f     =           g   S     ⁢     R   S   2         2   ⁢   π       ⁢            ⅆ     ω   0   ′         ⅆ     L   C                  ,           (   23   )             
 
 where  
           g   S     =       A   eff       λ   ⁢           ⁢   L         ,       
 
 A eff  is the effective tip area, and λ L  is the London penetration depth. Therefore, the sensitivity S f  for a superconductor is found by taking the derivative of ω′ 0  with respect to L C  in equation (22) and is given by  
               S   f     =           g   S     ⁢     R   S   2         2   ⁢   π       ⁢           ω   0     ⁡     (     1   -       C   C       2   ⁢     C   0           )       ⁡     [       1     (     2   ⁢     L   0       )       ⁢       C   C   2         (       C   C     +     C   0       )     2         ]       .               (   24   )             
 
         [0075]     The ability of the probe to differentiate between regions of different conductivity within a superconductor Δσ/σ is defined as  
               Δσ   σ     =         (       V     n   ⁡     (   rms   )           V     i   ⁢           ⁢   n         )     /     S   f       ⁢     S   r     ⁢     σ   .               (   25   )             
 
         [0076]     The probe couples to a metallic sample through the coupling capacitance C C  and the conductor is represented as the series combination of R S  and L S . An equivalent circuit of a metallic sample does not contain the circuit elements L C  and C S  in the two-fluid equivalent circuit (see  FIG. 13 ). Therefore, C S =0 and L C =∞. The impedance Z 1  is the series combination of C C , R S , and L S  and is represented as  
               Z   1     =         R   S     +     jω   ⁢           ⁢     L   S       +     1     jω   ⁢           ⁢     C   C           =         1   +     jω   ⁢           ⁢     C   C     ⁢     R   S       -       ω   2     ⁢     L   S     ⁢     C   C           jω   ⁢           ⁢     C   C         .               (   26   )             
 
         [0077]     The parallel combination of Z 1  and C 0  results in  
               1     Z   2       =       jω   ⁢           ⁢     C   0       +       jω   ⁢           ⁢     C   C           (     1   -       ω   2     ⁢     L   S     ⁢     C   C         )     +     jω   ⁢           ⁢     C   C     ⁢     R   S                         =         jω   ⁢           ⁢       C   0     ⁡     (     1   -       ω   2     ⁢     L   S     ⁢     C   C         )         +     jω   ⁢           ⁢     C   C       -       ω   2     ⁢     C   0     ⁢     C   C     ⁢     R   S             (     1   -       ω   2     ⁢     L   S     ⁢     C   C         )     +     jω   ⁢           ⁢     C   C     ⁢     R   S                         =       jω   ⁢           [       C   C     +       C   0     ⁡     (     1   -       ω   2     ⁢     L   S     ⁢     C   C         )       -       ω   2     ⁢     C   0     ⁢     C   C     ⁢     R   S         ]       1   -       ω   2     ⁢     L   S     ⁢     C   C       +     jω   ⁢           ⁢     C   C     ⁢     R   S             ,             
 
 and the impedance Z 2  is  
               Z   2     =           (     1   -       ω   2     ⁢     L   S     ⁢     C   C         )     +     jω   ⁢           ⁢     C   C     ⁢     R   S           jω   ⁡     [         C   C     ⁢       C   0     ⁡     (     1   -       ω   2     ⁢     L   S     ⁢     C   C         )         -       ω   2     ⁢     C   0     ⁢     C   C     ⁢     R   S         ]         =       -     j   ω       ⁢       Z   2   ′     .                 (   27   )             
 
         [0078]     The total impedance Z TOTAL  looking into the terminals of the probe coupled to a conductor sample is  
         Z   TOTAL     =       R   0     +     jω   ⁢           ⁢     L   0       -       j   ω     ⁢       Z   2   ′     .             
 
         [0079]     The complex impedance Z 3  can be represented as  
         Z   2     =         1   jω     ⁡     [     Re   ⁡     (     Z   2   ′     )       ]       =     -         j   ω     ⁡     [     Re   ⁡     (     Z   2   ′     )       ]       .             
 
         [0080]     At resonance, the inductive and capacitive reactance cancel; therefore,  
                   jω   ⁢           ⁢     L   0       -       j   ω     ⁡     [     Re   ⁢     (     Z   2   ′     )       ]         =   0     ,           ⁢         ω   2     ⁢     L   0       =     Re   ⁢       (     Z   2   ′     )     .                 (   28   )             
 
         [0081]     The impedance z′ 2  is represented as  
               Z   2   ′     =           (     1   -       ω   2     ⁢     L   S     ⁢     C   C         )     +     jω   ⁢           ⁢     C   C     ⁢     R   S           jω   ⁡     [       C   C     +       C   0     ⁡     (     1   -       ω   2     ⁢     L   S     ⁢     C   C         )       -       ω   2     ⁢     C   0     ⁢     C   C     ⁢     R   S         ]         .             (   29   )             
 
         [0082]     Taking the real part of equation (29), we have  
                     Re   ⁡     (     Z   2   ′     )       =           (     1   -       ω   2     ⁢     L   S     ⁢     C   C         )     ⁡     [       C   C     +       C   0     ⁡     (     1   -       ω   2     ⁢     L   S     ⁢     C   C         )         ]       +       ω   2     ⁢     C   0     ⁢     C   C   2     ⁢     R   S   2               [       C   C     +       C   0     ⁡     (     1   -       ω   2     ⁢     L   S     ⁢     C   C         )         ]     2     +       ω   2     ⁢     C   0     ⁢     C   C   2     ⁢     R   S   2                       =             C   C     ⁡     (     1   -       ω   2     ⁢     L   S     ⁢     C   C         )       +         C   0     ⁡     (     1   -       ω   2     ⁢     L   S     ⁢     C   C         )       2     +       ω   2     ⁢     C   0     ⁢     C   C   2     ⁢     R   S   2               [       C   C     +       C   0     ⁡     (     1   -       ω   2     ⁢     L   S     ⁢     C   C         )         ]     2     +       ω   2     ⁢     C   0     ⁢     C   C   2     ⁢     R   S   2           .                   (   30   )             
 
         [0083]     The numerator and denominator of equation (30) are considered separately, so the numerator is expanded and results in
 
(C C +C 0 )−ω 2 (L S C C   2 +2C 0 L S C C −C 0 C C   2 R S   2 )+ω 4 C 0 C C   2 L S   2 .  (31)
 
         [0084]     The ω 4  term in equation (31) is discarded due to insignificance and the denominator of equation (30) is expanded as
 
( C   C   +C   0 −ω 2   L   S   C   C   C   0 ) 2 +ω 2   C   0   2   C   C   2   R   S   2=(   C   C   +C   0 ) 2 −2ω 2   L   S ( C   C   +C   0 ) C   C   C   0 +ω 4   C   0   2   L   S   2 +ω 2   C   0   2   C   C   2   R   S   2 .  (32)
 
         [0085]     Likewise, the ω 4  term in equation (32) is neglected and the combination of equations (31) and (32) appear as  
                   (       C   C     +     C   0       )     -       ω   2     ⁡     (         L   S     ⁢     C   C   2       +     2   ⁢     C   0     ⁢     L   S     ⁢     C   C       -       C   0     ⁢     C   C   2     ⁢     R   S   2         )               (       C   C     +     C   0       )     2     -     2   ⁢     ω   2     ⁢       L   S     ⁡     (       C   C     +       C   0     ⁢     L   S         )       ⁢     C   0     ⁢     C   C       +       ω   2     ⁢     C   0   2     ⁢     C   C   2     ⁢     R   S   2           .           (   33   )             
 
         [0086]     Factoring out (C C +C 0 ) in numerator and denominator of equation (33) and substituting the result into equation (28) produces  
                 ω   2     ⁢     L   0       =       1     (       C   C     +     C   0       )       ⁢         1   -       ω   2     ⁢       (         L   S     ⁢     C   C   2       +     2   ⁢     C   0     ⁢     L   S     ⁢     C   C   2       -       C   0     ⁢     C   C   2     ⁢     R   S   2         )       (       C   C     +     C   0       )             1   -     2   ⁢     ω   2     ⁢         L   S     ⁢     C   C     ⁢     C   0         (       C   C     +     C   0       )         +       ω   2     ⁢         C   0   2     ⁢     C   C   2     ⁢     R   S   2           (       C   C     +     C   0       )     2             .               (   34   )             
 
         [0087]     Reducing equation (34) and multiplying by  
         1   +       ω   2     ⁢       [         L   S     ⁢       C   C     ⁡     (       C   C     +     2   ⁢     C   0         )         -       C   0     ⁢     C   C   2     ⁢     R   S   2         ]       (       C   C     +     C   0       )           ,       
 
 results in  
                 ω   2     ⁢     L   0       =       1     (       C   C     +     C   0       )       ⁢       1     1   +       ω   2     ⁢         L   S     ⁢     C   C   2         (       C   C     +     C   0       )         +       ω   2     ⁢         C   0     ⁢     C   C   2     ⁢     R   S   2         (       C   C     +     C   0       )       ⁢     (         C   0         C   C     +     C   0         -   1     )           .               (   35   )             
 
         [0088]     The relation ω 0   2 /(1+C C /C 0 ) with ω 0   2 =1/L 0 C 0  as a zero-order approximation to our iterative process is substituted into equation (35) producing a first-order approximation  
               ω   0   ′2     =       1       L   0     ⁡     (       C   C     +     C   0       )         ⁢       1     1   +         L   S       L   0       ⁢       (       C   C         C   C     +     C   0         )     2       -           C   0     ⁢     R   S   2         L   0       ⁢       (       C   C         C   C     +     C   0         )     3           .               (   36   )             
 
         [0089]     Rewriting equation (36) and taking the square root of both sides and neglecting higher-order terms, we have the first-order approximation for the perturbed resonant frequency due to the coupling of the probe to a conductor.  
               ω   0   ′     =       ω   0     ⁢     1       1   +       C   C       C   0             ⁢       1       1   +         L   S       L   0       ⁢       (       C   C         C   C     +     C   0         )     2             .               (   37   )             
 
         [0090]     The Taylor expansion of equation (37) gives  
               ω   0   ′     =           ω   0     ⁡     (     1   -       C   C       2   ⁢     C   0           )       ⁡     [     1   -         L   S       2   ⁢     L   0         ⁢       C   C   2         (       C   C     +     C   0       )     2           ]       .             (   38   )             
 
         [0091]     The sensitivity S f  for a conductor is defined as  
                 S   f     =           g   S     ⁢     R   S   2         2   ⁢   π       ⁢            ⅆ     ω   0   ′         ⅆ     L   S                  ,           (   39   )             
 
 where  
           g   S     =       A   eff     δ       ,       
 
 A eff  is the effective tip area, and δ is the skin depth. Therefore, the sensitivity S f  equation (39) for a conductor is found by taking the derivative of ω′ 0  with respect to L S  in equation (38) and results in  
               S   f     =           g   S     ⁢     R   S   2         2   ⁢   π       ⁢           ω   0     ⁡     (     1   -       C   C       2   ⁢     C   0           )       ⁡     [       1     2   ⁢     L   0         ⁢       C   C   2         (       C   C     +     C   0       )     2         ]       .               (   40   )             
 
         [0092]     The ability of the probe to differentiate between regions of different conductivity Δσ/σ is defined as  
                 Δσ   σ     =         (       V     n   ⁡     (   rms   )           V     i   ⁢           ⁢   n         )     /     S   f       ⁢     S   r     ⁢   σ       ,           (   31   )             
 
 where v n(rms)  is given in equation (11) and v in  is the probe input voltage. 
 
         [0093]     The probe also couples to a dielectric sample through the coupling capacitance C C  and the dielectric is represented as the parallel combination of R S  and C S . The equivalent circuit of an insulating sample does not contain the circuit elements L C  and L S  from the two-fluid equivalent circuit. Therefore, L S =0 and L C =∞. The impedance Z 1  is the parallel combination of R S  and C S  and is represented as  
               Z   1     =         R   S         jω   ⁢           ⁢     C   S     ⁢     R   S       +   1       .             (   42   )             
 
         [0094]     The series combination of Z 1  and C C  result in  
               Z   2     =         1     jω   ⁢           ⁢     C   C         +     1       jω   ⁢           ⁢     C   S     ⁢     R   S       +   1         =       1   +     jω   ⁢           ⁢     C   S     ⁢     R   S       +     jω   ⁢           ⁢     C   C     ⁢     R   S           jω   ⁢           ⁢       C   C     ⁡     (       jω   ⁢           ⁢     C   S     ⁢     R   S       +   1     )                     (   43   )             
 
         [0095]     The impedance Z 3  is the parallel combination of Z 2  and C 0  and is represented as  
                 1     Z   3       =         jω   ⁢           ⁢       C   C     ⁡     (     1   +     jω   ⁢           ⁢     C   S     ⁢     R   S         )                   ⁢     (     1   +     jω   ⁢           ⁢     C   C     ⁢     R   S       +     jω   ⁢           ⁢     C   S     ⁢     R   S         )         +     jω   ⁢           ⁢     C   0           ,     
     ⁢             Z             ⁢   3       =       1   +     jω   ⁢           ⁢     C   C     ⁢     R   S       +     jω   ⁢           ⁢     C   S     ⁢     R   S             jω   ⁢           ⁢       C   C     ⁡     (     1   +     jω   ⁢           ⁢     C   S     ⁢     R   S         )         +     jω   ⁢           ⁢       C   0     ⁡     (     1   +     jω   ⁢           ⁢     C   C     ⁢     R   S       +     jω   ⁢           ⁢     C   S     ⁢     R   S         )                         =       -     j             ⁢   ω         ⁢       Z             ⁢   3               ⁢   ′       .                       (   44   )             
 
         [0096]     The total impedance Z TOTAL  looking into the terminals of the probe coupled to a dielectric sample is  
         Z   TOTAL     =       R   0     +     jω   ⁢           ⁢     L   0       -       j   ω     ⁢       Z   3   ′     .             
 
         [0097]     The complex impedance Z 3  can be represented as  
         Z   3     =         1   jω     ⁡     [     Re   ⁡     (     Z   3   ′     )       ]       =     -         j   ω     ⁡     [     Re   ⁡     (     Z   3   ′     )       ]       .             
 
         [0098]     At resonance, the inductive and capacitive reactance cancel; hence,  
                   jω   ⁢           ⁢     L   0       -       j   ω     ⁡     [     Re   ⁢     (     Z   3   ′     )       ]         =   0     ,           ⁢         ω   2     ⁢           ⁢     L   0       =     Re   ⁢       (     Z   3   ′     )     .                 (   45   )             
 
         [0099]     The quantity jωR S  is factored out in the numerator and denominator of equation (44) and the result is placed into equation (45), giving  
                 ω   2     ⁢     L   0       =     Re   ⁢     {       1   +     jω   ⁢           ⁢       R   S     ⁡     (       C   C     +     C   S       )               (       C   C     +     C   0       )     +     jω   ⁢           ⁢       R   S     ⁡     [         C   C     ⁢     C   S       +       C   0     ⁡     (       C   C     +     C   S       )         ]             }                   =           (       C   C     +     C   0       )     +       ω   2     ⁢         R   S   2     ⁡     (       C   C     +     C   S       )       ⁡     [         C   C     ⁢     C   S       +       C   0     ⁡     (       C   C     +     C   S       )         ]                 (       C   C     +     C   S       )     2     +         ω   2     ⁡     [         C   C     ⁢     C   S       +       C   0     ⁡     (       C   C     +     C   S       )         ]       2         .               
 
         [0100]     R S  is neglected since it is large, so  
             ω   2     ⁢     L   0       ≈       (       C   C     +     C   S       )           C   C     ⁢     C   S       +       C   0     ⁡     (       C   C     +     C   S       )             =       1     C   0       ⁢       1     [     1   +         C   C     ⁢     C   S           C   0     ⁡     (       C   C     +     C   S       )           ]       .           
 
 Therefore,  
               ω   0     ′   2       =       1       L   0     ⁢     C   0         ⁢       1     [     1   +         C   C     ⁢     C   S           C   0     ⁡     (       C   C     +     C   S       )           ]       .               (   46   )             
 
         [0101]     Solving for ω′ 0  in equation (46) results in  
               ω   0   ′     =       ω   0     ⁢       1       1   +         C   C     ⁢     C   S           C   0     ⁡     (       C   C     +     C   S       )               .               (   47   )             
 
         [0102]     The Taylor expansion of equation (47) gives  
               ω   0   ′     =         ω   0     ⁡     [     1   -         C   C     ⁢     C   S         2   ⁢       C   0     ⁡     (       C   C     +     C   S       )             ]       .             (   48   )             
 
         [0103]     The sensitivity S f  for a dielectric is defined as  
                 S   f     =         g   S       2   ⁢   π       ⁢            ⅆ     ω   0   ′         ⅆ     C   S                  ,           (   49   )             
 
 where  
           g   S     =       A   eff       ξ   S         ,       
 
 A eff  is the effective tip area, and ξ S  is the decay length of the evanescent wave, which is approximately 100 μm. Therefore, the sensitivity S f  for a dielectric is found by taking the derivative of ω′ 0  with respect to C S  in equation (48)  
               S   f     =           g   S     ⁢     ω   0         4   ⁢   π       ⁢         C   C   2           C   0     ⁡     (       C   C     +     C   S       )       2       .               (   50   )             
 
         [0104]     The ability of the probe to differentiate between regions of different permittivity Δ∈/∈ is defined as  
                 Δ   ⁢           ⁢   ɛ     ɛ     =         (       V     n   ⁡     (   rms   )           V     i   ⁢           ⁢   n         )     /     S   f       ⁢     S   r     ⁢     ɛ   .               (   51   )             
 
         [0105]     The experimental verification of the sensitivity for superconductors is performed on a YBa 2 Cu 3 O 7-δ  coated SrTiO 3  bi-crystal of 6° orientation mismatch. Resonant frequency shift measurements are taken, resulting in complex permittivity values for two separate locations below T c  at 79.4 K. The measurements are taken in the boundary at points C and D shown in  FIG. 8 . The sensitivities given by equations (14), (24), and (25) are listed in Table II.  
                                                           TABLE II                           SENSITIVITY AND ASSOCIATED PARAMETERS       FOR SUPERCONDUCTORS                ε′/ε 0  (10 8 )   S r     S f     Δσ/σ                        Position C   −8.94   9.03 × 10 −6     1.13 × 10 −6     1.0 × 10 −2         Position D   −8.87   1.04 × 10 −5     1.13 × 10 −6     8.6 × 10 −3                    
 
         [0106]     The sensitivity parameters comprise C C =1.36×10 −15  F, C 0 =8.91×10 −12  F, L 0 =2.03×10 −8  H, R S =1×10 −6  Ω, σ=3.3×10 8  S/m, and g s =1.02×10 −3 . The experimental results show that Δσ/σ≅7.8×10 −3 .  
         [0107]     The experimental verification of the sensitivity for conductors is also performed on the YBa 2 Cu 3 O 7-δ  coated SrTiO 3  bi-crystal of 6° orientation mismatch. The measurements are taken at the same locations for the superconductor sensitivity, in the boundary at points C and D ( FIG. 14 ) at a temperature of 300 K. The sensitivities given by equations (14), (40), and (41) are listed in Table III. The sensitivity parameters consist of C C =1.36×10 −15  F, C 0 =8.91×10 −12  F, L 0 =2.03×10 −8  H, R S =7.76×10 −4  Ω[8], σ=1.28×10 3  S/m, and g c =1.54×10 −4 . The experimental results have shown that Δσ/σ≅2.4×10 −2 .  
                                                           TABLE III                           SENSITIVITY AND ASSOCIATED       PARAMETERS FOR CONDUCTORS                ε″/ε 0     S r     S f     Δσ/σ                        Position C   6.3   6.83 × 10 −6     5.9 × 10 −2     8.36 × 10 −2         Position D   6.15   5.95 × 10 −6     5.9 × 10 −2     9.91 × 10 −2                    
 
         [0108]     The experimental verification of the sensitivity for dielectrics is performed on single crystal SrTiO 3  utilizing the ferroelectric dependence on temperature property of the material, i.e., ∈ r =f(T). The probe tip is set to a 1 μm distance above the sample and tuned to a resonant frequency of 1.114787 GHz at a temperature of 300 K and is illustrated in  FIG. 15 . The temperature is raised in 0.2 K increments until the resonance shifted in frequency to 1.114792 GHz at 302 K due to the change in dielectric constant and is shown in  FIG. 16 . The change in dielectric constant is determined using the Curie-Weiss law and results in Δ∈/∈≅6.23×10 −3 . The sensitivity parameters consist of ∈′/∈ 0 =320.8, C C =1.36×10 −15  F, C 0 =8.91×10 −12  F, C S =4.37×10 −15  F, and g s =1.54×10 −6 . The lowest theoretically estimated change in permittivity that can be detected by the sensor was Δ∈/∈=5.75×10 −4 .  
         [0109]     It is noted that terms like “preferably,” “commonly,” and “typically” are not utilized herein to limit the scope of the claimed invention or to imply that certain features are critical, essential, or even important to the structure or function of the claimed invention. Rather, these terms are merely intended to highlight alternative or additional features that may or may not be utilized in a particular embodiment of the present invention.  
         [0110]     Having described the invention in detail and by reference to specific embodiments thereof, it will be apparent that modifications and variations are possible without departing from the scope of the invention defined in the appended claims. Accordingly, all variations of the present invention that would readily occur to one of ordinary skill in the art are contemplated to be within the scope of the present invention. Thus, the present invention is not to be limited to the examples and embodiments set forth herein. Rather, the claims alone shall set for the metes and bounds of the present invention.