Patent Publication Number: US-8526105-B2

Title: Structural illumination and evanescent coupling for the extension of imaging interfermetric microscopy

Description:
RELATED APPLICATIONS 
     This application is a divisional of U.S. patent application Ser. No. 12/347,619 filed Dec. 31, 2008, now U.S. Pat. No. 8,115,992 issued on Feb. 14, 2012, and claims priority from U.S. Provisional Patent Application Ser. Nos. 61/017,985, filed Dec. 31, 2007; 61/089,669, filed Aug. 18, 2008; and 61/115,246, filed Nov. 17, 2008, which are hereby incorporated by reference in their entirety. 
    
    
     GOVERNMENT RIGHTS 
     This invention was made with government support under Contract Nos. HR0011-05-1-0006 awarded by the Defense Advanced Research Projects Agency and FA9550-06-1-0001 awarded by the Air Force Office of Scientific Research. The government has certain rights in the invention. 
    
    
     FIELD OF THE INVENTION 
     This invention relates generally to microscopy, and, more particularly, to an imaging interferometric microscope; coherent anti-Stokes Raman (CARS) microscope; methods for applying the imaging interferometric microscopy to coherent anti-Stokes Raman (CARS) microscopy; methods to simplify the approaches to imaging interferometric microscopy that require less, or no, access to the optical path after the objective lens and so can be applied to existing microscopes; and for providing an optical resolution up to about λ\4n, where n is the substrate refractive index and λ is an optical wavelength. 
     BACKGROUND OF THE INVENTION 
     Optical microscopy is among the oldest applications of optical science and remains one of the most widely used optical technologies. In spite of impressive results obtained by fluorescent microscopy in exceeding the classical diffraction limit, non-fluorescent transmission/reflection microscopy remains an important field of modern research. However, using traditional illumination schemes, resolution is limited to ˜K 1 λ/NA where λ is the source wavelength and NA is the numerical aperture (sine of the half-acceptance angle) of the imaging objective lens. The “constant” K 1  depends on both the details of the image and on the illumination scheme. Hence, traditional approaches to improve resolution are either to use shorter wavelengths and/or to use larger numerical-aperture lenses. For biological samples, however, the wavelength is constrained to the visible spectral range because ultraviolet photons can damage samples. In many practical cases, even for inorganic samples, the wavelength is limited to the deep ultraviolet (for example 193 nm) since transmissive optical materials become difficult at shorter wavelengths (fused quartz has a cutoff at ˜185 nm). Furthermore, a disadvantage of using a high-NA lens is the resulting short depth-of-field (an essential feature of achieving high resolution in a single image; typically the depth-of-field scales as K 2 λ/NA 2  where K 2  is a second “constant” of order unity). The depth-of-field decreases rapidly as the NA is increased to increase the resolution. In addition, the field of view (the area over which the resolution is achieved) and the working distance (the distance from the final lens surface to the object plane) are reduced for higher-NA optical systems. These latter two issues can be surmounted by more complex objective lenses, with an increase in the cost of manufacturing. These tradeoffs are well known and are discussed in many microscopy overviews. 
     Synthetic aperture approaches, such as, for example, imaging interferometric microscopy (IIM) extend the collected spatial frequencies to improve the image. IIM uses a low-NA objective and yet provides a resolution approximately a factor of two better than that available even with a high-NA objective using conventional coherent or incoherent illumination. A major advantage is that the depth-of-field, field-of-view and working distance associated with the low-NA system are retained, but the final composite image has a resolution at the linear system limit imposed by the transmission medium (≧λ/4 where λ is the wavelength in the transmission medium), and significantly better than that accessible with even a high NA lens using conventional (coherent or incoherent) illumination approaches. As is well-known, using off-axis illumination provides enhanced resolution over that available with either of the standard illumination schemes discussed above, but there is some distortion of the image associated with the resultant non-constant transfer function for different regions of frequency space. This non-uniform frequency-space coverage can be addressed with appropriate pupil plane filters and by combining partial images corresponding to different parts of frequency space, as has been previously demonstrated in the case of imaging interferometric lithography. 
     An exemplary IIM with two offset partial images, one each in orthogonal spatial directions can result in an increased resolution by three times using about 0.4-NA objective and 633-nm He—Ne laser. Furthermore, IIM requires building an interferometric system around the objective lens which is an issue for wide-spread adoption of this approach, and in particular towards its adoption to the existing microscopes. In the prior art, this interferometer required additional optics to relay the pupil plane of the collection objective to convenient location; this is straightforward but required significant additional optics. Hence, there is a need for a new approach that does not require a large change to the imaging optical system that comprises the objective lens and subsequent optical components. 
     The prior art imaging interferometric microscopy was able to image maximum spatial frequency of 2π/λ e.g. to the linear system&#39;s limit of the air (transmission medium between the object and the lens). The ultimate linear system&#39;s limit is 2πn/λ, which reflects the use of an immersion medium of refractive index n. Even though materials with refractive indices of upto about 3.3 are known at some optical wavelengths, the highest numerical aperture available for the immersion microscopy is about 1.4, limited by the refractive index of the glass used to make the lens, by the refractive indices available for the index matching fluids, and the well known difficulties of making aberration corrected optics of high NA. Hence, there is a need for a new approach that can achieve this linear system&#39;s limit without requiring high index matching fluids or high NA lenses. 
     SUMMARY OF THE INVENTION 
     In accordance with various embodiments, an apparatus for coherent anti-Stokes Raman (CARS) microscopy is disclosed. The apparatus can comprises an object plane on which is disposed a first surface of a planar substrate, wherein the substrate is characterized by a homogeneous refractive index and a surface normal, wherein the substrate is arranged to support an object; a first optical system disposed to provide an illumination of the object plane, the illumination characterized by two substantially coincident coherent beams with wavelengths λ 1  and λ 2  and corresponding angular frequencies ω 1  and ω 2  with ω 1 &gt;ω 2 , a radius of curvature, and disposed at one of a plurality of incident wave vectors from about 0 to about 2πn sub /λ 1  with respect to the surface normal of the substrate and at a plurality of azimuth angles spanning about 0 to about 2π; a second optical system having an optical axis disposed at one of a plurality of center wave vectors from about 0 to about 2πn sub /λ 1  with respect to the surface normal of the substrate, wherein the second optical system is characterized by a numerical aperture and is responsive to optical signals at frequencies greater than ω 1 ; a third optical system disposed between the first optical system and an entrance aperture of a first optical element of the second optical system to provide interferometric reintroduction of the illumination as a reference beam at an angular frequency of 2ω 1 −ω 2  into the second optical system, wherein each of an amplitude, a phase, a radius of curvature and an angle of incidence of the reference beam is arranged to be adjusted such that a corrected reference wave is present at an image plane of the second optical system; an electronic image device disposed at the image plane of the second optical system and operable to respond linearly to a local optical intensity and transfer a local optical intensity map across the image plane as a sub-image to a signal processor device in electronic form; one or more devices operable to adjust the first, the second, and the third optical systems to collect sub-images for different pairs of the pluralities of incident illumination wave vectors from the first optical system and collection angles from the second optical system so as to sequentially obtain a plurality of sub-images corresponding to a plurality of regions of spatial frequency space, wherein the signal processor device is operable to sequentially receive the electronic form of the sub-images and manipulate the sub-images to correct for distortions and alterations introduced by the optical configuration, store, and combine the plurality of sub-images corresponding to the plurality of regions of spatial frequency space to create a composite image. In some aspects, the substrate can be air. 
     In some aspects, the third optical system can further comprise a first beamsplitter disposed in an optical path of the first optical system before the object to collect a portion of the coherent illumination; and one or more optics disposed between the first optical system and an entrance aperture of a first optical element of the second optical system, wherein the optics comprises a nonresonant nonlinear material configured to generate anti-Stokes four-wave mixing frequency 2ω 1 −ω 2  and exclude the frequencies ω 1  and ω 2  and to interferometrically reintroduce a portion of the anti-Stokes illumination as the reference beam into the second optical system in a position after an exit aperture of a collection lens of the second optical system, wherein the reintroduction is at one of a position corresponding to a position a zero-order beam would have had if it had been transmitted through an appropriate higher numeric aperture lens of the second optical system or an aliased position to reduce pixel requirements of the electronic image device, wherein the signal processor is operable to compensate for spatial frequency aliasing. 
     In some aspects, the third optical system can further comprise a first beamsplitter disposed in an optical path of the first optical system to collect a portion of the coherent illumination; one or more transfer optics disposed between the first optical system and an entrance aperture of a first optical element of the second optical system, wherein the optics comprises a nonresonant nonlinear material configured to generate anti-Stokes four-wave mixing frequency 2ω 1 −ω 2  and exclude the frequencies ω 1  and ω 2 ; and a second beamsplitter disposed between the object and an entrance aperture of a collection lens of the second optical system to reintroduce the portion of the anti-Stokes coherent illumination as the reference beam into the second optical system at an angle less than an entrance angular aperture of the second optical system. 
     In some aspects, the third optical system can further comprise a first beamsplitter disposed in an optical path of the first optical system to collect a portion of the coherent illumination; one or more transfer optics disposed between the first optical system and the second optical system, wherein the optics comprises a nonresonant nonlinear material configured to generate anti-Stokes four-wave mixing frequency 2ω 1 −ω 2  and exclude the frequencies ω 1  and ω 2 ; and at least one of a grating or a grating on a waveguide disposed between the object and an entrance aperture of a first optical element of the second optical system to reintroduce a portion of the anti-Stokes coherent illumination as the reference beam into the second optical system at an angle less than an entrance angular aperture of the second optical system. 
     In some aspects, the third optical system can further comprise a first beamsplitter disposed in an optical path of the first optical system to collect a portion of the coherent illumination; one or more transfer optics, wherein the optics comprises a nonresonant nonlinear material configured to generate anti-Stokes four-wave mixing frequency 2ω 1 −ω 2  and exclude the frequencies ω 1  and ω 2 ; an optical element operable to direct a portion of the anti-Stokes coherent illumination as the reference beam to illuminate the object at an angle corresponding to less than an entrance angular aperture of the second optical system; and a dynamic physical block disposed in a back pupil plane of the second optical system to alternately block and unblock a portion of a pupil aperture corresponding to the position of the reference beam in the aperture. 
     In some aspects, the third optical system can further comprise a first beamsplitter disposed in an optical path of the first optical system to collect a portion of the coherent illumination; one or more transfer optics, wherein the optics comprises a nonresonant nonlinear material configured to generate anti-Stokes four-wave mixing frequency 2ω 1 −ω 2  and exclude the frequencies ω 1  and ω 2 ; an optical element operable to direct the portion of the anti-Stokes coherent illumination as the reference beam to illuminate the object at an angle corresponding to less than an entrance angular aperture of the second optical system; and a guided-mode resonance filter disposed between the object and a collection lens of the second optical system to sequentially block and unblock the transmission of the reference beam. 
     In some aspects, the apparatus can further comprise at least one known reference object to cover a part of an image field. 
     In some aspects, the first, the second, and the third optical systems are arranged in at least one of a transmission configuration or a reflection configuration. 
     In some aspects, the plurality of incident wave vectors of the first optical system can comprise wave vectors &lt;2π/λ 1 , wherein the wave vectors are accessed by illumination of the substrate at polar angles between 0 and π/2. 
     In some aspects, the plurality of incident wave vectors of the first optical system can comprise wave vectors between 2π/λ 1  and 2πn sub /λ 1 , wherein the wave vectors are accessed by evanescent wave illumination of the object through the substrate of refractive index n sub . 
     In some aspects, the plurality of center wave vectors of the second optical system can comprise only center wave vectors &lt;2π/λ 1 , wherein the center wave vectors are accessed by an optical system above the object plane of the substrate. 
     In some aspects, the plurality of center wave vectors of the second optical system can comprise center wave vectors between 2π/λ 1  and 2πn sub /λ 1 , wherein the center wave vectors greater than 2π/λ 1  are accessed through the substrate and the second optical system comprises a plurality of gratings on a side of the substrate opposite the object plane, wherein each grating is characterized by a position, a pitch, and a grating profile. 
     In accordance with some aspects of the present disclosure, a method for coherent anti-Stokes Raman (CARS) microscopy is disclosed. The method can comprise providing an object atop an object plane disposed upon a planar substrate, wherein the substrate is characterized by a homogeneous refractive index (n sub ) and a surface normal; providing a first optical system disposed to provide an illumination of the object plane, the illumination characterized by two substantially coincident coherent beams with wavelengths λ 1  and λ 2  and corresponding angular frequencies ω 1  and ω 2  with ω 1 &gt;ω 2 , a radius of curvature, and disposed at one of a plurality of incident wave vectors from about 0 to about 2πn sub /λ 1  with respect to the surface normal of the substrate and at a plurality of azimuth angles spanning 0 to about 2π; providing a second optical system having an optical axis disposed at one of a plurality of center wave vectors from about 0 to about 2πn sub /λ 1  with respect to the surface normal, wherein the second optical system is characterized by a numerical aperture and is responsive to optical signals at frequencies greater than ω 1 ; providing a third optical system disposed between the first optical system and an entrance aperture of a first optical element of the second optical system to provide interferometric reintroduction of a reference illumination as a reference beam at a frequency of 2ω 1 −ω 2 , into the second optical system, wherein each of an amplitude, a phase, a radius of curvature and an angle of incidence of the reference is operable to be adjusted such that a corrected reference wave is present at an image plane of the second optical system; recording a sub-image of the object at an object plane using an electronic image device disposed at the image plane of the second optical system that responds linearly to a local optical intensity and transfers a local optical intensity map across the image plane as a sub-image to a signal processor in electronic form, wherein the sub-image is formed as a result of interference between the scattering resulting from the coherent illumination of the object and the reference beam; providing one or more devices operable to adjust the first, the second, and the third optical systems to sequentially collect sub-images for different pairs of the pluralities of illumination wave vectors from the first optical system and collection angles from the second optical system so as to sequentially obtain a plurality of sub-images corresponding to a plurality of regions of spatial frequency space; providing a signal processor device operable to sequentially receive the electronic form of the sub-images and manipulate the sub-images to correct for distortions and alterations introduced by the optical configuration, to store and combine the plurality of sub-images corresponding to the plurality of regions of spatial frequency space into a composite image. 
     In some aspects, the method can further comprise tuning a frequency difference ω 1 −ω 2  of the two substantially coplanar and spatially coherent plane waves through Raman resonances of one or more materials in the object. 
     In some aspects, the substrate can be air. 
     In some aspects, the method can include providing a third optical system further comprising collecting a portion of the coherent illumination using a first beamsplitter disposed in an optical path of the first optical system; and interferometrically reintroducing using one or more optics disposed between the first optical system and an entrance aperture of a first optical element of the second optical system a coherent anti-Stokes 2ω 1 −ω 2  reference beam and excluding the frequencies ω 1  and ω 2  into the second optical system in a position after an exit aperture of a collection lens, wherein the reintroduction is at one of a position corresponding to a position a zero-order beam would have had if it had been transmitted through an appropriate numeric aperture lens of the second optical system or an aliased position to reduce pixel requirements of the electronic image device, wherein the signal processor is adjusted to compensate for this spatial frequency aliasing. 
     In some aspects, the method can include providing a third optical system further comprising providing a first beamsplitter in an optical path of the first optical system to collect a portion of the coherent illumination; using one or more transfer optics disposed between the first optical system and the second optical system, wherein the optics comprises a nonresonant nonlinear material configured to generate anti-Stokes four-wave mixing frequency 2ω 1 −ω 2  and exclude the frequencies ω 1  and ω 2 ; and interferometrically injecting an anti-Stokes reference beam using a second beamsplitter disposed between the object and a collection lens of the second optical system at an angle less than an entrance angular aperture of the second optical system. 
     In some aspects, the method can include providing the third optical system further comprising providing a first beamsplitter in an optical path of the first optical system to collect a portion of the coherent illumination; providing one or more transfer optics disposed between the first optical system and the second optical system, wherein the optics comprises a nonresonant nonlinear material configured to generate anti-Stokes four-wave mixing frequency 2ω 1 −ω 2  and exclude the frequencies ω 1  and ω 2 ; and using at least one of a grating or a grating on a waveguide disposed between the object and a collection lens of the second optical system to reintroduce the portion of the anti-Stokes coherent illumination as the reference beam into the second optical system at an angle less than an entrance angular aperture of the second collection optical system. 
     In some aspects, the method can include providing the third optical system further comprising providing a first beamsplitter in an optical path of the first optical system to collect a portion of the coherent illumination; using one or more transfer optics, wherein the optics comprises a nonresonant nonlinear material configured to generate anti-Stokes four-wave mixing frequency 2ω 1 −ω 2  and exclude the frequencies ω 1  and ω 2  to direct the portion of the anti-Stokes coherent illumination as the reference beam to illuminate the object at an angle corresponding to less than an entrance angular aperture of the second optical system; and providing a dynamic physical block disposed in a back pupil plane of the second optical system to alternately block and unblock a portion of a pupil aperture corresponding to a position of the reference beam in the aperture. 
     In some aspects, the method can include providing the third optical system further comprising providing a first beamsplitter in an optical path of the first optical system to collect a portion of the coherent illumination; providing one or more transfer optics, wherein the optics comprises a nonresonant nonlinear material configured to generate anti-Stokes four-wave mixing frequency 2ω 1 −ω 2  and exclude the frequencies ω 1  and ω 2 ; directing the portion of the anti-Stokes coherent illumination as the reference beam to illuminate the object at an angle corresponding to less than an entrance angular aperture of the second optical system; and providing a guided-mode resonance filter between the object and a collection lens of the second optical system to sequentially block and unblock transmission of the reference beam. 
     In some aspects, the method can include providing at least one known reference object to cover a part of an image field. 
     In some aspects, the first, the second, and the third optical systems can be arranged in at least one of a transmission configuration or a reflection configuration. 
     In some aspects, the plurality of incident wave vectors of the first optical system can comprise only wave vectors &lt;2π/λ 1 , wherein the wave vectors are accessed by illumination of the substrate at polar angles between 0 and π/2. 
     In some aspects, the plurality of incident wave vectors of the first optical system can comprise wave vectors between 2π/λ 1  and 2πn sub /λ 1 , wherein the wave vectors are accessed by evanescent wave illumination of the object through the substrate. 
     In some aspects, the plurality of center wave vectors of the second optical system can comprise only center wave vectors &lt;2π/λ 1 , wherein the center wave vectors are accessed by an optical system above the object plane of the substrate. 
     In some aspects, the plurality of center wave vectors of the second optical system can comprise center wave vectors between 2π/λ 1  and 2πn sub /λ 1 , wherein the center wave vectors greater than 2π/λ 1  are accessed through the substrate and the second optical system comprises a plurality of gratings on the side of the planar substrate opposite the object plane, wherein each grating is characterized by a position, a pitch, and a grating profile. 
     In some aspects, the method can further comprise one or more processes of subtracting dark field images, subtracting of background images, shifting of spatial frequencies in accordance with an optical configuration, and eliminating one or more overlapping coverage of frequency space. 
     In some aspects, the method can further comprise selecting regions of spatial frequency space to provide the image of the object in the object plane. 
     Additional objects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. The objects and advantages of the invention will be realized and attained by means of the elements and combinations particularly pointed out in the appended claims. 
     It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the invention, as claimed. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate several embodiments of the invention and together with the description, serve to explain the principles of the invention. 
         FIG. 1  shows an exemplary prior art imaging interferometric microscopy (IIM) experimental arrangement. 
         FIG. 2A  shows the frequency space coverage for conventional normal incidence coherent illumination. 
         FIG. 2B  shows the frequency space coverage for an off-axis incidence coherent illumination. 
         FIG. 3  shows an exemplary structured illumination approach to IIM, according to various embodiments of the present teachings. 
         FIG. 4  shows the schematic of structural illumination and restoration algorithms, in accordance with the present teachings. 
         FIG. 5A  is a schematic illustration showing a dynamic physical block in a back pupil plane of the second optical system to alternately block and unblock the reference beam, according to present teachings. 
         FIG. 5B  is a schematic illustration showing injection of the reference beam into a second optical system using a prism, according to present teachings. 
         FIG. 5C  is a schematic illustration of injection of the reference beam into a second optical system using a beamsplitter, according to present teachings. 
         FIG. 5D  is a schematic illustration showing blocking of the reference beam with a k-vector filter, according to present teachings. 
         FIG. 5E  is a schematic illustration showing injection of the reference beam with a grating, according to present teachings. 
         FIG. 6  shows k-vector filter characteristic of an exemplary SiN-on-glass guided mode resonance filter. 
         FIG. 7A  shows the frequency space coverage for the arrangement of  FIG. 3 , with intermediate frequency offset within bandpass of lens. 
         FIG. 7B  shows the frequency space coverage for the arrangement of  FIG. 3  after removal of the dark field and intermediate frequency imaging terms and correction of the observed frequencies. 
         FIG. 8A  schematically illustrates a Manhattan geometry pattern used for image resolution exploration consisting of five nested “ells” and a large box. 
         FIG. 8B  illustrates intensity Fourier space components of the Manhattan geometry pattern shown in  FIG. 8A , mapped onto a frequency space coverage of the imaging system. 
         FIGS. 9A-9F  show the preliminary results of an experiment using an NA=0.4 objective with a He—Ne laser illumination (λ=633 nm) and with about 240 nm structure with corresponding simulations using the configuration presented in  FIG. 5A . 
         FIG. 10A  shows reconstructed images of 260 nm and 240 nm CD structures obtained using the optical configuration of  FIG. 5A  after the dark field subtraction, frequency shifting correction, and sub-image combination. 
         FIG. 10B  show a crosscut (gray) of the images of  FIG. 10A  compared with a crosscut of corresponding simulation (black). 
         FIG. 11A  shows reconstructed images of 260 nm and 240 nm CD structures obtained using the optical arrangement shown in  FIG. 5E . 
         FIG. 11B  shows a crosscut (gray) of the images of  FIG. 10A  compared with a crosscut of corresponding simulation (black). 
         FIG. 12A  shows an exemplary IIM arrangement with evanescent illumination, according to various embodiments of the present teachings. 
         FIG. 12B  shows an exemplary IIM arrangement with evanescent illumination with a rotated optical axis, according to various embodiments of the present teachings. 
         FIGS. 13A-13C  show alternate approaches for coupling light for substrate illumination, in accordance with various embodiments. 
         FIG. 14A  schematically illustrates Manhattan geometry pattern used for image resolution exploration consisting of five nested “ells” and a large box. 
         FIG. 14B  illustrates intensity Fourier space components of the Manhattan geometry pattern shown in  FIG. 14A , mapped onto a frequency space coverage of the imaging system, for a structure with CD=180 nm for the optical arrangement shown in  FIG. 3 . 
         FIG. 14C  illustrates intensity Fourier space components of the Manhattan geometry pattern shown in  FIG. 12A , mapped onto a frequency space coverage of the imaging system, for a structure with CD=150 nm for the optical arrangement shown in  FIG. 11A . 
         FIG. 15A  show reconstructed image of 260 nm and 240 nm CD structures obtained using the optical arrangement shown in  FIG. 12A . 
         FIG. 15B  show a crosscut (gray) of the images of  FIG. 15A  compared with a crosscut of corresponding simulation (black). 
         FIG. 16A  shows a reconstructed high frequency image of a 150 nm structure using evanescent illumination and a tilted optical system, shown in  FIG. 12B . 
         FIG. 16B  shows a high frequency image simulation of a 150 nm structure using evanescent illumination and a tilted optical system, shown in  FIG. 12B . 
         FIG. 16C  shows experimental and simulation cross-cuts of images shown in  FIGS. 16A and 16B . 
         FIG. 16D  shows a reconstructed composite image of a 150 nm structure using evanescent illumination and a tilted optical system, shown in  FIG. 12B   
         FIG. 16E  shows a composite image simulation of a 150 nm structure using evanescent illumination and a tilted optical system, shown in  FIG. 12B . 
         FIG. 16F  shows experimental and simulation cross-cuts of images shown in  FIGS. 16D and 16E . 
         FIG. 17  shows available frequency space coverage for various IIM optical configurations, in accordance with the present teachings. 
         FIG. 18  shows a schematic diagram showing the high angle light scattered from an object and extracted from the substrate using at least one grating, in accordance with various embodiments of the present teachings. 
         FIG. 19  shows prism coupling for extracting light scattered into a substrate, in accordance with various teachings. 
         FIGS. 20A and 20B  show embodiments for tiling frequency space with a substrate (n=1.5) in one direction, in accordance with various embodiments. 
         FIG. 21  show an exemplary tiling with almost complete frequency space coverage (NA=0.65, n=1.5), in accordance with present teachings. 
         FIG. 22  show another exemplary tiling with a larger NA objective lens (NA=0.95, n=1.5), in accordance with various embodiments. 
         FIG. 23  show another exemplary tiling strategy for high index substrate (n=3.6, collection NA=0.65, as in  FIG. 18 ), in accordance with various embodiments. 
     
    
    
     DESCRIPTION OF THE EMBODIMENTS 
     Reference will now be made in detail to the present embodiments, examples of which are illustrated in the accompanying drawings. Wherever possible, the same reference numbers will be used throughout the drawings to refer to the same or like parts. 
     Notwithstanding that the numerical ranges and parameters setting forth the broad scope of the invention are approximations, the numerical values set forth in the specific examples are reported as precisely as possible. Any numerical value, however, inherently contains certain errors necessarily resulting from the standard deviation found in their respective testing measurements. Moreover, all ranges disclosed herein are to be understood to encompass any and all sub-ranges subsumed therein. For example, a range of “less than 10” can include any and all sub-ranges between (and including) the minimum value of zero and the maximum value of 10, that is, any and all sub-ranges having a minimum value of equal to or greater than zero and a maximum value of equal to or less than 10, e.g., 1 to 5. In certain cases, the numerical values as stated for the parameter can take on negative values. In this case, the example value of range stated as “less that 10” can assume negative values, e.g. −1, −2, −3, −10, −20, −30, etc. 
       FIG. 1  shows a prior art imaging interferometric microscopy (IIM) arrangement  100 . As shown in  FIG. 1 , a collimated (equivalent to coherent) illumination beam  110  is incident on an object  120  at an angle of incidence θ. In the illustrated case, θ is beyond the collection angle of the objective lens  130  and an auxiliary optical system  135  is shown schematically to collect the zero-order transmission  109 , appropriately adjust its divergence, direction, and phase and re-inject it onto the image plane  124  where it interferes with the diffracted beams  101 ,  102 ,  103 ,  104  from the object  120  to construct a partial image. Alternatively, instead of using the zero-order transmission  109 , which might be blocked by the objective-lens  130  mount, a portion of the illumination beam  110  can be split off before the object  120  and directed around the objective lens  130 . The interference between the zero-order beam  109  and the diffracted beams  101 ,  102 ,  103 ,  104  transmitted through the objective lens  130  can shift the collected diffracted information back to high frequency. As a result of the square-law intensity response, the resulting frequency coverage  140 B can be represented by a pair of circles  144 ,  146  of radius NA/λ shifted away from zero frequency  142  by 2(2π)NA/λ as shown in  FIG. 2B .  FIG. 2B  shows the frequency space coverage  140 B for off-axis coherent illumination, where frequencies beyond the lens bandpass are recorded in the sub-image as a result of the interferometric reconstruction. For comparison,  FIG. 2A  shows the frequency space coverage  140 A for conventional normal incidence on-axis coherent illumination. 
     An object of the present teachings is to reduce or eliminate the requirement of the prior art for optical access to between the back of the objective lens and the image plane of the second optical system. This access is required for injecting the reference beam  109  in the prior art ( FIG. 1 ). However, in many existing optical microscopes, this region is inaccessible. The structured illumination approach disclosed herein provides alternative methods of injecting the reference beam in front of the objective lens of the second optical system, thereby simplifying the application of imaging interferometric microscopy to existing optical microscopy systems. Since both the diffracted beams and the reference beams are not transmitted through the same objective, characterized by an NA, the high frequency image components are necessarily shifted to lower frequency. This is similar to the use of an intermediate frequency in heterodyne radio receivers, but in the spatial frequency rather than the temporal frequency domain. It is necessary to reset the spatial frequencies by signal processing after each sub-image is captured in the electronic recording device. Additional advantages of this approach are that the pixel count requirements in the image plane are reduced, since only lower spatial frequencies, up to 2(2π)NA/λ, are recorded, and the interferometer can be made smaller since all of the components are on the front side of the objective lens, reducing vibrational effects on the interferometric reconstruction. 
       FIG. 3  shows an optical arrangement of the apparatus  200  for an exemplary structured illumination approach to IIM, according to various embodiments of the present teachings. The apparatus  200  can include an object  220  disposed on an object plane  222  on which a first surface of a substrate  225  can be disposed, wherein the substrate  225  can be characterized by a homogeneous refractive index (n sub ) and a surface normal  226 . The apparatus  200  can also include a first optical system including a source  211  and one or more optical components (not shown) disposed to provide a substantially coherent illumination  210  of the object plane  222 , the illumination  210  characterized by a wavelength λ and a radius of curvature and disposed at one of a plurality of incident wave vectors from about 0 to about 2π/λ with respect to a surface normal of the substrate and at a plurality of azimuth angles spanning 0 to 2π. Illumination  210  is diffracted by object  220  into a central, undiffracted 0 th  order  209  and higher diffraction orders including a 1 st  order  201 , a 2 nd  order  202 , a 3 rd  order  203  and a 4 th  order  204 . Each of these orders corresponds to a different spatial frequency component of the object, with smaller spatial frequencies corresponding to higher diffraction angles (orders). The apparatus  200  can also include a second optical system  230  disposed to collect portions of the illumination beam scattered from the object plane  222 , the second optical system having an optical axis  236  disposed at one of a plurality of center wave vectors from about 0 to about 2π/λ with respect to the substrate surface normal  226  and at the azimuth angle corresponding to the illumination of the first optical system, wherein the second optical system  230  is characterized by a numerical aperture (NA). In various embodiments, the second optical system  230  can include at least one objective lens. The apparatus  200  can also include a third optical system represented by beamsplitter  205  and mirror  206  disposed between the optical path of the first optical system and an entrance aperture of the second optical system to provide interferometric reintroduction of a portion of the coherent illumination (reference beam)  210 ′ into the second optical system  230 , wherein each of an amplitude, a phase, a radius of curvature, a path length and an angle of incidence of the reference beam can be adjusted such that a correct reference beam can be present at a image plane  224  of the second optical system  230 . It is understood that additional optical components not shown are necessary to achieve this correct reference beam. The reference beam  210 ′ is obtained by splitting off a portion of the illumination beam  210  with beamsplitter  205  and redirecting the split-off beam with third optical system and the apparatus  200  is configured so that the total path lengths of the illumination beam  210  and the reference beam  210 ′ from the beam splitter to the image plane  224  are within the temporal coherence length of the source to insure interferometric reconstruction of the sub-image. Illumination  210 ′ is diffracted by substrate  225  into a central, undiffracted 0 th  order  209 ′ and higher diffraction orders including a 1 st  order  201 ′ and a 2 nd  order  202 ′. The apparatus can also include an electronic image device  228  disposed at the image plane  224  of the second optical system  230  that responds linearly to the local optical intensity and transfers the local optical intensity map across the image plane (a sub-image) to a signal processor device  260  in electronic form. In various embodiments, the electronic image device  228  can be a charged coupled device (CCD) camera, a CMOS (complementary metal-oxide semiconductor) camera, and any similar electronic focal plane array device. The apparatus  200  can further include a device  250  for adjusting the first, the second, and the third optical systems to collect sub-images for different pairs of the pluralities of incident (first optical system) and collection center (second optical system) wave vectors so as to sequentially obtain a plurality of sub-images corresponding to a plurality of regions of spatial frequency space. In various embodiments, the device can block/unblock various beams, rotate substrate etc. In some embodiments, the device can include mechanical components, such as, for example, motors. In other embodiments, the device can include electronic components, such as, for example, acoustic modulators or similar devices. The signal processor device  260  can also be arranged to sequentially receive the electronic form of the sub-images and manipulate the sub-images to correct for distortions and alterations introduced by the optical configuration, store, and combine the plurality of sub-images corresponding to the plurality of regions of spatial frequency space to create a composite image. In some other embodiments, the signal processor device can include one or more computers. In some embodiments, the first, the second, and the third optical systems can be arranged in a transmission configuration with respect to the substrate surface. In other embodiments, the first, the second, and the third optical systems can be arranged in a reflection configuration with respect to the substrate surface. Items  250 ,  235  and  270  represent means to alter the various optical systems to correspond to different angles of incidence and scattering as described below. 
     In certain embodiments apparatus  200  for an exemplary structured illumination approach to IIM can also include at least one known reference object to cover a small part of the image field. 
     According to various embodiments, there is a method for structural imaging interferometric microscopy. The method can include providing an object  220  disposed over a planar substrate  225 , wherein the substrate  225  is characterized by a homogeneous refractive index (n sub ) and a surface normal  226  and providing a first optical system to illuminate the object  220  with substantially coherent illumination  210 , the illumination characterized by a wavelength λ and a radius of curvature and disposed at one of a plurality of incident wave vectors from about 0 to about 2π/λ with respect to a surface normal of the substrate and at a multiplicity of azimuth angles spanning from about 0 to about 2π. The method can also include providing a second optical system  230  disposed to collect portions of the illumination scattered from the object plane  222 , the second optical system  230  having an optical axis  236  disposed at one of a plurality of center wave vectors from about 0 to about 2π/λ with respect to the substrate  225  surface normal  226  and at the azimuth angle corresponding to the illumination of the first optical system, wherein the second optical system  230  is disposed such that the object  220  is substantially at the object plane  222  of the second optical system  230  which is characterized by a numerical aperture (NA). The method can further include providing a third optical system disposed between the optical path of the first optical system and an entrance aperture of the second optical system to provide interferometric reintroduction of a portion of the coherent illumination (reference beam)  210 ′ into the second optical system, wherein each of an amplitude, a phase, a radius of curvature and an angle of incidence of the reference can be adjusted such that a corrected reference wave is present at the image plane of the second optical system, wherein the corrected reference beam  210 ′ and the illumination beam  210  are within the temporal coherence length of the source. The method can also include recording a sub-image of the object  220  at an object plane  222  using an electronic image device  228 , wherein the sub-image is formed as a result of interference between the scattering resulting from the coherent illumination of the object  220  and the reference beam  210 ′. The method can also include adjusting the first, the second, and the third optical systems to sequentially collect a plurality of sub-images corresponding to a plurality of regions of spatial frequency space, manipulating each of the plurality of sub-images using a signal processor to correct for distortions and alterations introduced by the optical configuration, and combining the plurality of sub-images into a composite image to provide a substantially faithful image of the object  220 . In various embodiments, the method can further include one or more processes of subtraction of dark field images, subtraction of background images, shifting of spatial frequencies in accordance with the optical configuration, and elimination of one or more overlapping coverages of the frequency space wherein the elimination operations can be performed either in the optical systems or in the signal processing. In some embodiments, the method can also include selecting the regions of spatial frequency space to provide a more or less faithful image of the object  220  in the object plane  222 . One of ordinary skill in the art would know that the regions of frequency space that are important vary depending on the object. For example for a Manhattan geometry pattern, there is less need to gather spectral information on the diagonals. See, for example, Neumann et al. in Optics Express, Vol. 16, No. 10, 2008 pp 6785-6793 which describes a structural illumination for the extension of imaging interferometric microscopy, the disclosure of which is incorporated by reference herein in its entirety. 
       FIG. 4  shows a flow diagram schematic, indicated generally by  400 ( a ), ( b ) and ( c ), of structural illumination and restoration algorithms. High spatial frequencies represented by diffracted beams from the off-axis illumination are mixed with the local oscillator beam, the dark field of the Image is subtracted as is the low frequency image without its dark field. Then the spatial frequencies are reset in Fourier space and the whole image is reconstructed by combining high and low frequency sub-images. 
     To mathematically explain the structured illumination approach to IIM, first a simple mathematical description of a conventional coherent illumination microscopy image will be described and then the mathematical description will be extended to the prior art IIM experiment and finally to the structured illumination approach. 
     The total transmission through an arbitrary object (assumed to be periodic on large scale to allow Fourier sums rather than Fourier integrals) and illuminated by a plane wave at normal incidence can be given by: 
                       ∑       ∀   k     ,     l   ∈   R         ⁢       A     k   ,   l       ⁢     exp   ⁡     [       ⅈ   ⁢           ⁢   x   ⁢           ⁢   k   ⁢           ⁢     ω   x       +     ⅈ   ⁢           ⁢   y   ⁢           ⁢   l   ⁢           ⁢     ω   y         ]       ⁢     ⅇ     ⅈ   ⁢           ⁢     γ     k   ,   l       ⁢   z           =         A     0   ,   0       ⁢     ⅇ     ⅈ   ⁢           ⁢     γ     0   ,   0       ⁢   z         +       ∑     k   ,     l   ≠   0         ⁢       A     k   ,   l       ⁢     exp   ⁡     [       ⅈ   ⁢           ⁢   xk   ⁢           ⁢     ω   x       +     ⅈ   ⁢           ⁢   yl   ⁢           ⁢     ω   y         ]       ⁢     ⅇ     ⅈ   ⁢           ⁢     γ     k   ,   l       ⁢   z                     (   1   )               
where ω x , ω y  are the discrete spatial frequency increments of the Fourier summation; x and y are orthogonal spatial coordinates;
 
               γ     k   ,   l       ≡           (       2   ⁢   π   ⁢           ⁢   n     λ     )     2     -       (     k   ⁢           ⁢     ω   x       )     2     -       (     l   ⁢           ⁢     ω   y       )     2               
with n the refractive index of the transmission medium (1 for air); R is the set of integers, for which (|γ k,l |) 2 &gt;0, that is the range of integers for which the diffracted beams are within the bandpass of the transmission medium and are propagating in the z-direction, away from the object. Note that this representation is a scalar approximation that is appropriate as long as the angles do not get too large, and it is assumed below that all beams are polarized in the same direction. A more rigorous treatment is straightforward, but mathematically gets more complex and obscures the physical insight in these simpler equations.
 
     The transmission through the optical system adds a filter factor: 
     
       
         
           
             
               
                 
                   
                     
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                   2 
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     The transmission function of the objective lens is a simple bandpass function: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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                   3 
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     and the final image intensity can be obtained by taking the square modulus of equation 2, viz (shown in Appendix A): 
     APPENDIX A 
     
       
         
           
             
               
                 
                   
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                                       ( 
                                       
                                         
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                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Each of the three lines in this result has a simple physical interpretation. The top line is a constant independent of spatial coordinates, equal to the average intensity of the pattern. This ensures that the intensity is always positive as physically required. The second line represents the imaging terms that we want to retain. Finally the third line is the cross-correlation of the diffracted beams with themselves equivalent to the dark field image that would be obtained if the zero-order diffraction (transmission) was blocked at the back pupil plane. The imaging terms are band-limited to transverse spatial frequencies of (2π/λ)NA; the dark field terms extend out to (4π/λ)NA and are typically weaker in intensity than the imaging terms since for an object with thickness &lt;&lt;λ, |A 0,0 | is larger than any of the diffracted terms. In all of the summations the summation indices extend over all terms in R except for the zero-order term which has been explicitly separated. Equation 4 gives the intensity over all space beyond the objective lens. The image is obtained in the back image plane (z=0) where the exponentials in γz vanish. The focusing information is contained in these exponential terms and its characteristic length, the depth-of-field, depends on the NA, as is well known. A Fourier optics perspective provides additional insight into the three terms. The DC term (top line) is a δ-function at the origin. The image terms fill a circle of radius 2πNA/λ as a result of the band-limited transmission function. Finally, the dark-field image contains frequencies up to 4πNA/λ as a result of the interference of the various diffracted orders. 
     It is well-known that additional, higher spatial frequency, information can be accessed with off-axis illumination.  FIG. 1  shows a conventional IIM arrangement  100 , wherein a collimated illumination beam  110  can be incident on an object  120  at an angle of incidence θ. In particular in the case of IIM, the offset angle is chosen such that the zero-order transmission (reflection) is beyond the lens ( 130 ) NA, 
               ω   offset     &gt;         2   ⁢   π   ⁢           ⁢   NA     λ     .           
The result is that higher spatial frequency information is transmitted through the lens, but only a dark field image is recorded in a traditional coherent illumination microscopy configuration (without the reference beam  109 ). This is solved in IIM by introducing an auxiliary optical system  135 , an interferometer that reinjects the zero-order transmission on the low-NA side of the lens to reset the spatial frequencies. In practice it is simpler to reintroduce the zero-order transmission as an appropriately mode matched point source in the back pupil plane without actually using the transmitted beam which is often blocked by the objective lens mount. Effectively, the interferometer results in a modified filter transfer function where the zero-order is transmitted even though it is outside the lens NA. The amplitude, the phase, and the offset position in the back focal plane of the objective have to be controlled to provide a correct sub-image. These are often set by using a nearby, known reference object along with the object of interest.
 
     It is straightforward to extend the mathematical treatment to the off-axis illumination case. Equation 2 can be modified to (shown in Appendix B): 
     APPENDIX B 
     
       
         
           
             
               
                 
                   
                     
                       A 
                       
                         0 
                         , 
                         0 
                       
                       ′ 
                     
                     ⁢ 
                     
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                           - 
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                         ⁢ 
                         
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                           off 
                         
                         ⁢ 
                         x 
                       
                     
                     ⁢ 
                     
                       ⅇ 
                       
                         
                           ⅈ 
                           ⁡ 
                           
                             ( 
                             
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                                 , 
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                               ′ 
                             
                             ) 
                           
                         
                         ⁢ 
                         z 
                       
                     
                   
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                         , 
                         
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                           ≠ 
                           0 
                         
                       
                     
                     ⁢ 
                     
                       
                         T 
                         ⁡ 
                         
                           ( 
                           
                             
                               
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                                 ⁢ 
                                 
                                     
                                 
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                                 off 
                               
                             
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                       ⁢ 
                       
                         exp 
                         ⁡ 
                         
                           [ 
                           
                             
                               ⅈ 
                               ⁢ 
                               
                                   
                               
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                                 x 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       k 
                                       ⁢ 
                                       
                                           
                                       
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                                       off 
                                     
                                   
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                                 y 
                               
                               ⁢ 
                               y 
                             
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           ⅈ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
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                               k 
                               , 
                               l 
                             
                             ′ 
                           
                           ⁢ 
                           z 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     where ω off =2π sin (θ off )/λ is the frequency offset arising from the off-axis illumination at angle θ off  (assumed in the x-direction), the primes on the γs indicate that the propagation directions take into account the offset illumination, and the prime on the A 0,0  refers to the re-injected 0-order. 
     Taking the square of equation 5 can provide the intensity on the imaging camera (shown in Appendix C): 
     APPENDIX C 
                      A     0   ,   0     ′          2     +     …   ⁢           ⁢     (     dc   ⁢           ⁢   offset     )                       ∑     k   ,     l   ≠   0         ⁢       A     0   ,   0     ′     ⁢     A     k   ,   l     *     ⁢     T   ⁡     (         k   ⁢           ⁢     ω   x       -     ω   off       ;     l   ⁢           ⁢     ω   y         )       ⁢     exp   ⁡     [       ⅈ   ⁢           ⁢   k   ⁢           ⁢     ω   x     ⁢   x     +     ⅈ   ⁢           ⁢   l   ⁢           ⁢     ω   y     ⁢   y       ]       ⁢     ⅇ       ⅈ   ⁡     (       γ     0   ,   0     ′     -     γ     k   ,   l         )       ⁢   z           +       c   .   c   .     +   …       ⁢           ⁢     (   imaging   )     ⁢     
     ⁢       ∑     k   ,     l   ≠   0         ⁢       ∑               ′     ,     k   ′     ,       l   ′     ≠   0         ⁢       A     k   ,   l       ⁢     T   ⁡     (         k   ⁢           ⁢     ω   x       -     ω   off       ;     l   ⁢           ⁢     ω   y         )       ⁢     A       n   ′     ,     l   ′       *     ⁢     T   ⁡     (           k   ′     ⁢     ω   x       -     ω   off       ;       l   ′     ⁢     ω   y         )       ⁢     exp   ⁡     [         ⅈ   ⁡     (     k   -     k   ′       )       ⁢     ω   x     ⁢   x     +       ⅈ   ⁡     (     l   -     l   ′       )       ⁢     ω   y     ⁢   y       ]       ⁢     ⅇ       ⅈ   ⁡     (       γ     k   ,   l     ′     -     γ       k   ′     ,     l   ′       ′       )       ⁢   z       ⁢           ⁢   …   ⁢           ⁢     (     dark   ⁢           ⁢   field     )                   
where the three terms on separate lines correspond to (top) a constant term, (middle) the imaging terms and (bottom) the dark field image. Subtracting out the dark field terms (by taking an image with the interferometer blocked so that only the third term survives) provides a sub-image that accurately captures the spatial frequency components that are transmitted through the optical system. Note that the imaging terms (middle line) are at the correct frequencies and that the offset illumination angle has cancelled out of the expression except for the filter transmission functions.
 
     Changing both the illumination angle (and the angle of reintroduction) and the azimuthal angle changes the offset allowing recording of a different region of frequency space. Specifically, for Manhattan geometry (x,y oriented patterns) a second offset exposure to capture the high spatial frequencies in the y-direction, that is with the substrate rotated by π/2, can be used. Additional spatial frequency terms can be captured with large illumination angles. 
     Referring back to the  FIG. 3 , in the exemplary structured illumination approach to IIM, there can be two coherent illumination beams  210 ,  210 ′, the first beam  210  can be at the same offset as in the previous example so that ω offset  is &gt;NA/λ, and the second beam  210 ′ can be at the maximum offset that fits through the lens ω off ≦NA/λ, denoted as ω NA  in the equation. Then the fields are: 
                       A     0   ,   0       ⁢     exp   ⁡     (       -   i     ⁢           ⁢     ω   off     ⁢   x     )       ⁢     ⅇ     i   ⁢           ⁢     γ     0   ,   0     off     ⁢   z         +       ∑     k   ,     l   ≠   0         ⁢       A     k   ,   i       ⁢     exp   ⁡     [         i   ⁡     (       k   ⁢           ⁢     ω   x       -     ω   off       )       ⁢   x     +     il   ⁢           ⁢     ω   y     ⁢   y       ]       ⁢     ⅇ     i   ⁢           ⁢     γ     k   ,   l     off     ⁢   z           +       B     0   ,   0       ⁢     exp   ⁡     (       -   i     ⁢           ⁢     ω   NA     ⁢   x     )       ⁢     ⅇ     i   ⁢           ⁢     γ     0   ,   0     NA           +       ∑     p   ,     r   ≠   0         ⁢       B     p   ,   r       ⁢     exp   ⁡     [         i   ⁡     (       p   ⁢           ⁢     ω   x       -     ω   NA       )       ⁢   x     +     ir   ⁢           ⁢     ω   y     ⁢   y       ]       ⁢     ⅇ     i   ⁢           ⁢     γ     p   ,   r     NA     ⁢   z                   (   7   )               
where the series with coefficients A k,j  are due to the first offset beam ( 210 ) and the second series with the coefficients B p,q  are due to the second offset beam ( 210 ′) and squaring while taking advantage of the fact that without the interferometer the A 0,0  beam  209  is not transmitted to the objective image plane while the B 0,0  beam  209 ′ is transmitted through the lens  230  gives (equation 8 shown in Appendix D):
 
     APPENDIX D 
     
       
         
           
             
               
                 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                  
                                 
                                   B 
                                   
                                     0 
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                               2 
                             
                             + 
                             
                               
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                                     ≠ 
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                               ⁢ 
                               
                                 
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                                 ⁢ 
                                 
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                                     r 
                                   
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                                 ⁢ 
                                 
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                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         
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                                       ; 
                                       
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                                           y 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 exp 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     ⅈ 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
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                                       ; 
                                       
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                     } 
                   
                   + 
                 
               
               
                 
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                   I 
                   ] 
                 
               
             
             
               
                 
                   
                     { 
                     
                       
                         
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                                     off 
                                   
                                 
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                                             NA 
                                           
                                         
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                       + 
                       
                         c 
                         . 
                         c 
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                     } 
                   
                   + 
                 
               
               
                 
                   [ 
                   II 
                   ] 
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           
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                             , 
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                         ⁢ 
                         
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                               l 
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                                   off 
                                 
                               
                               ; 
                               
                                 l 
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                           exp 
                           ⁡ 
                           
                             [ 
                             
                               
                                 
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                                     ( 
                                     
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                         ⁢ 
                         
                           ⅇ 
                           
                             
                               ⅈ 
                               ⁡ 
                               
                                 ( 
                                 
                                   
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                                       k 
                                       , 
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                                     γ 
                                     
                                       
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                                         l 
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                   + 
                   
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                     . 
                     c 
                     . 
                   
                 
               
               
                 
                   [ 
                   III 
                   ] 
                 
               
             
             
               
                 
                   
                     
                       ∑ 
                       
                         k 
                         , 
                         l 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           p 
                           , 
                           
                             r 
                             ≠ 
                             0 
                           
                         
                       
                       ⁢ 
                       
                         
                           A 
                           
                             k 
                             , 
                             l 
                           
                         
                         ⁢ 
                         
                           B 
                           
                             p 
                             , 
                             r 
                           
                           * 
                         
                         ⁢ 
                         
                           T 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   k 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ω 
                                     x 
                                   
                                 
                                 - 
                                 
                                   ω 
                                   off 
                                 
                               
                               ; 
                               
                                 l 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ω 
                                   y 
                                 
                               
                             
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                         ⁢ 
                         
                           T 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   p 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
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                                 - 
                                 
                                   ω 
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                               ; 
                               
                                 r 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ω 
                                   y 
                                 
                               
                             
                             ) 
                           
                         
                         × 
                         
                           exp 
                           [ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 ⅈ 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     k 
                                     - 
                                     p 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
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                                 x 
                               
                             
                             + 
                             
                               
                                 ⅈ 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ω 
                                       NA 
                                     
                                     - 
                                     
                                       ω 
                                       off 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
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                                 ⁡ 
                                 
                                   ( 
                                   
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                                     - 
                                     r 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
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                                 y 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               ⅈ 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     γ 
                                     
                                       k 
                                       , 
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                                     off 
                                   
                                   - 
                                   
                                     γ 
                                     
                                       p 
                                       , 
                                       
                                         r 
                                         ′ 
                                       
                                     
                                     NA 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             z 
                           
                         
                       
                     
                   
                   + 
                   
                     c 
                     . 
                     c 
                     . 
                   
                 
               
               
                 
                   [ 
                   IV 
                   ] 
                 
               
             
           
         
       
     
     The first three terms in the upper bracket, labeled [I], in equation 8 are the result of the off-axis illumination at the edge of the pupil. This image can be measured independently by blocking the extreme off axis beam and subtracted from the result. The term labeled [II] is the desired information, the image terms beating against a zero-order beam; because the zero-order beam is not at the correct angle to reset the frequencies to match the object frequencies (adjusted for magnification) there is a shift between the observed and the actual image plane frequencies {exp[i(ω NA −ω off ) x ]} that will need to be fixed computationally (e.g. one is measuring the Fourier components at an intermediate frequency as detailed above). [III] is the dark field from the extreme off-axis illumination. Finally the last term, [IV] is the cross-correlated dark field from the two illumination beams. 
     To remove the unwanted terms in equation 8, five strategies can be used. However, these are not intended to be all-inclusive and other possibilities may exist. These are illustrated schematically in  FIGS. 5A-5E . There are two general approaches, in the  FIGS. 5A-5C , the reference beam is added before the object plane. This adds to some additional complexity in that the off axis and the reference beams give rise to diffracted information and it is necessary to separate out the information corresponding to the diffraction from the reference beam from the off axis beam. This can be accomplished as shown in the scheme outlined in  FIG. 4 . In  FIGS. 5D and 5E , the reference beam is added after the object plane and before the entrance to the collection lens. In these configurations, the reference beam does not illuminate the object and hence there is no additional diffraction. This simplifies the analysis, but at the cost of adding additional optical components in the region of limited access. 
       FIG. 5A  shows the first embodiment, wherein the third optical system  500 A can further include a first beamsplitter disposed in the optical path of the first optical system to collect a portion of the coherent illumination, one or more optical components to direct the portion of the coherent illumination as a reference beam  510 ′ ( 509 ′ on the second optical system  530  side of the object  520 ) to illuminate the object  520  at an angle θ corresponding to less than the entrance angular aperture (&lt;˜sin −1  NA) of the second optical system  530 , and a dynamic (adjustable on/off) physical block  550  disposed in a back pupil plane of the second optical system  530  to alternately block and unblock a small portion of the pupil aperture corresponding to the position of the reference beam  510  in the aperture. One of the advantages of this embodiment is that all of the information can be retained. However, this embodiment requires access to the illumination system pupil. In the case shown in  FIG. 5A , the objective pupil has been relayed to an auxiliary plane where it might be easier to access. The details of this optical configuration will depend on the optical construction of the objective lens. 
       FIG. 5B  shows the second embodiment  500 B, wherein both illumination beams can be shifted slightly using a prism  560  so that the zero order  209 ′ can be blocked but there is no change in the exponential factor, only in the transmission factors. Using the first and second embodiments, one can obtain and subtract dark field optical intensities from the image formed by interference of low and high frequencies (the second, fourth, and fifth terms of equation 6). Then one can subtract low frequency image without dark field and restore high frequency image by shifting frequencies in Fourier space. The second embodiment can be implemented easily and does not require any access to the objective pupil plane but it has some image-dependent information loss associate with the shifting of the illumination angles. As shown in  FIG. 5B , the prism is located in between the object  520  and the entrance aperture of the objective lens  530 ; alternatively it can be located before the object  520 . Depending on the specifics of the object  520 , it may be advantageous to dither the position of only the reference zero-order beam or of both zero-order beams. 
       FIG. 5C  shows yet another embodiment using a guided-mode filter (k-vector filter)  582  to block the zero order transmission just before the objective  530  and transmit the diffracted information at all other angles.  FIG. 6  shows an exemplary experimental un-optimized k-vector filter characteristic of a silicon-nitride-on-glass guided mode resonance filter, with a narrow angular width of the coupling. U.S. Pat. No. 5,216,680 discloses guided mode filter which can be used as an optical filter with very narrow line width and as an efficient optical switch, the disclosure of which is incorporated by reference herein in its entirety. Referring back to  FIG. 5C , it is possible to switch the zero-order on and off by mechanical dithering of the angular position or by dithering by a small degree of rotation around the optical axis, shown generally by  590 . This will allow identification of the source of the diffracted waves in the sub-image. Accordingly, the exemplary third optical system  500 C of the apparatus  200  in accordance with various embodiments, can further include one or more optical components to direct the portion of the coherent illumination as a reference beam to illuminate the object  520  at an angle θ less than the entrance angular aperture (&lt;˜sin −1  NA) of said second collection optical system  530 , the guided-mode resonance filter (k-vector filter)  582  disposed between the object plane  522  and a collection lens of the second optical system  530 , and a device (not shown) to adjust the position, tilt and rotation of the guided-mode resonance filter  582  between positions, tilts and rotations in which it alternately transmits and blocks the portion of the reference beam transmitted through the object plane. 
       FIG. 5D  shows yet another exemplary third optical system  500 D of the apparatus  200  in accordance with various embodiments. The third optical system  500 D can further include a first beamsplitter disposed in the optical path of the first optical system to collect a portion of the coherent illumination, one or more transfer optics disposed between the first optical system and the second optical system, and at least one of a grating  584  or a grating on a waveguide disposed between the object plane  522  and a front aperture of the collection lens (objective) of the second optical system  530  to reintroduce the portion of the coherent illumination as a reference beam into the second optical system  530  at an angle θ less than the entrance angular aperture (&lt;˜sin −1  NA) of the second optical system. In various embodiments, the grating  584  can have a short pitch (high spatial frequency) to avoid diffraction of the waves incident onto the grating  584  into new directions that are captured by the objective lens of the second optical system  530 . A major advantage of this method is that it does not require switchable gratings or mechanical actuation of the filter, since modulation is by simple blocking of the incident beam. 
       FIG. 5E  shows the fifth embodiment, wherein the zero order  510 ′ can be re-injected after the object and just in front of the objective by a beamsplitter  570 . Accordingly, the third optical system can include a second beamsplitter  570  disposed between the object plane  522  and a front aperture of a collection lens (objective) of the second optical system  530  to reintroduce the portion of the coherent illumination as a reference beam  510 ′ into the second optical system  530  at an angle θ less than the entrance angular aperture (&lt;˜sin −1  NA) of the second optical system  530 . The beamsplitter  570  can eliminate all of the diffracted beams associated with the local oscillator, B p,r =0, ∀p, r≠0, and simplifies equation 8. The third embodiment does not contain the first, second and fifth terms at all, so it is the most robust for the image processing, but images can be distorted by aberration caused by the beamsplitter; this aberration can be corrected with additional optical components. Using a very thin beamsplitter, e.g., a pellicle, eliminates much of the aberration associated with an expanding beam passing through a tilted thin plate. The use of a beamsplitter impacts the working distance of the objective, but the depth-of-field and field-of-view advantages of IIM are retained. 
       FIGS. 7A and 7B  show the frequency space coverage for the structured illumination approach to IIM shown in  FIG. 3  and using the first embodiment as shown in  FIG. 5A . All of the recorded frequencies are within the bandpass of the objective lens. The two offset circles  741 ,  742  in  FIG. 7A  correspond to coverage  740 A of both the intermediate frequency imaging terms and the offset frequency imaging terms beating with the intermediate frequency local oscillator.  FIG. 7B  shows the coverage  7408  after the unwanted dark field and local oscillator self-imaging terms have been removed and the spatial frequencies have been reset to their correct values. 
       FIGS. 8A and 8B  illustrate the frequency space coverage of partial images.  FIG. 8A  shows an illustration of a Manhattan (x, y) geometry pattern  800 A used for image resolution exploration consisting of five nested “ells” and a large box. The lines and spaces of the “ells” are about 240 nm.  FIG. 8B , indicated generally at  800 B, shows the intensity Fourier space components of the pattern  800 A, mapped onto the frequency space coverage of the imaging system using a NA=0.4 objective and an illumination wavelength of 633 nm (HeNe laser source). The resolution limit of this microscope system with conventional illumination is ˜0.6λ/NA (˜950 nm). The two circles at radii of NA/λ (0.4/λ) and 2NA/λ (0.8/λ) correspond to the approximate frequency space limits for coherent and incoherent illumination, respectively, and reflect the low-pass transmission characteristic of the objective lens. The inner sets of small shifted circles (radius NA/λ) in  FIG. 8B , that extend from −3NA/λ to 3NA/λ (±1.2/λ) in the x- and y-directions, show the frequency space coverage added with two offset partial images, one in each direction. The imaging is single side-band, only the diffracted plane waves to one side of the object are collected (opposite to the tilt of the illumination beam), the square law (intensity) response of the image formation and detection process restores the conjugate frequency space components, resulting in the two symmetrically displaced circles in  FIG. 8B  for each partial image. The offset (off-axis tilt) for these images was chosen at 2(2π)NA/λ to ensure that there was no overlap between the spectral coverage of the low-frequency partial image (extending out to NA/λ) and the offset images. As discussed previously, improved images can be obtained by subtracting the dark-field components of the image (with the zero-order transmission blocked). In the present embodiments, this provided a cosmetic, not a dramatic, improvement to the images. Additional frequency space coverage is available with a second pair of off-axis images, represented by the outer sets of shifted circles, with a larger tilt of the illumination plane wave, approaching grazing incidence (limited to 80° by practical considerations such as Fresnel reflectivity in the present experiment). The maximum frequency coverage in these images extends to [sin(80)+NA]/λ=(0.98+NA)/λ=(1.38/λ). The frequency-space coverage of the outer circles may be necessary to capture the fundamental frequency components of the line-space portion of this pattern. There is significant overlap between the frequency coverage of the first and second set of partial images as illustrated in  FIG. 8B . To provide a faithful image, the double coverage of frequency space associated with the image spectral overlaps can be excluded. This can be accomplished by filtering the images either optically (with appropriate apertures in the back focal plane) or electronically once the images are recorded. Importantly, since each of the partial images involves only the NA of the objective, this imaging concept retains the working distance, depth-of-field and field-of-view associated with the low-NA objective, but has a resolution beyond that achievable with even the highest NA objective using traditional illumination approaches. 
       FIGS. 9A-9F  show the preliminary results of an experiment using an NA=0.4 objective with a He—Ne laser illumination (λ=633 nm) and with about a 240 nm critical dimension structure with corresponding simulations using the configuration presented in  FIG. 5D , blocking the zero-order beam of the reference in the objective lens pupil.  FIG. 9A  is the mixed image corresponding to the interference of the low and high images and  FIG. 9B  is the corresponding simulation result.  FIG. 9C  is the image after subtraction dark field and low frequency image and  FIG. 9D  is the corresponding simulation result.  FIG. 9E  is restored high frequency image and  FIG. 9F  is the corresponding simulated result. 
     Similarly, results using dynamic (adjustable on/off) physical block presented in  FIG. 5A  are shown in  FIG. 10A . The same 260- and 240-nm objects are imaged as in  FIG. 9C ; only the final results after the dark field subtraction, frequency shifting correction and sub-image combination are shown. The corresponding cross-cuts are shown in  FIG. 10B . A total of four offset images, two each in the x- and y-directions, with θ ill =53° and 80° were used along with a 0.4 NA objective. As discussed previously, this configuration provided resolution to &lt;˜240 nm CD. There is overlap in the frequency space coverage between these two exposures and electronic frequency space filtering is used to assure a uniform coverage of frequency space. The present Manhattan geometry structure has spectral content concentrated along the x- and y-directions, so the offset illuminations were restricted to those directions. Adding additional frequency-space coverage for arbitrarily shaped structures can be accomplished by taking additional sub-images with rotation of the object in the (x,y) plane. The spatial frequency content of the image covers a wide range as a result of the large box (at 10× the linewidth of the line:space structures). The reconstructed image of the same structures obtained by the method with the beamsplitter configuration presented in  FIG. 5E  is shown in  FIG. 11A  and a crosscut of the image with corresponding simulation is shown in  FIG. 11B . The quality of the results for both methods is quite comparable. The second method retains a long working distance, but requires access to the imaging system pupil for blocking the zero-order. The first method does not require any modification to the traditional microscopy components, but has reduced working distance due to the beamsplitter in front of the objective. There are some extra features experimentally as compared to the model due to the lack of precision in mutual phase determination between the sub-images and speckle effects from the coherent illumination. These issues can be reduced by using improved arrangements and lower coherence sources. There are other possible alternatives; the optimum choice will depend on the specifics of the object and the constraints of specific optical systems. 
     The embodiments discussed so far provide spatial frequency coverage up to 2π(sin(θ ill )+NA)/λ≲2π(1+NA)/λ; that is the maximum illumination angle offset can be set close to 90° (providing the “1”) and the maximum angle collected by the objective lens corresponds to sin −1  (NA). As was previously disclosed in relation to the interferometric implementation of IIM, additional spatial frequencies are available by tilting the object plane relative to the objective lens axis. This allows collection of spatial frequencies up to 4π/λ, independent of the lens NA. The cost is a more complex signal processing requirement since the tilted object plane results in a nonlinear mapping of spatial frequencies from the object plane to the laboratory image that must be corrected to achieve a good image. This mapping has been discussed previously. The additional frequency space (and hence smaller image features) are available in the structured illumination embodiments of IIM disclosed herein. 
     Immersion microscopy is well known to provide higher spatial frequencies by a factor of the refractive index of the immersion medium, thereby extending the spatial frequency range to as high as 2n/λ. Again the advantages of immersion are directly applicable to structured illumination IIM. 
     Traditionally immersion microcoscopy has been practiced in reflection with a liquid medium on top of the object, or in transmission where advantage is taken of the high refractive index of the substrate (n sub ) as well as that of a liquid on top of the object. An intermediate possibility is to use just the refractive index of the substrate without an immersion fluid. In this case the spatial frequency range is extended to 2π(n sub +NA)λ.  FIG. 12  shows an exemplary apparatus  1200  for microscopy with an IIM arrangement with illumination by evanescent waves extending from a substrate, where 12A and B refers to alternate positions of the collection lens, according to various embodiments of the present teachings. The apparatus  1200  can include an object plane  1222  on which can be disposed a first surface of a substrate  1225 , wherein the substrate  1225 , upon which is an object  1220 , is characterized by a homogeneous refractive index (n sub ) and a surface normal  1226 . The apparatus  1200  can also include a first optical system, including prisms  1262  and substrate  1265 , disposed to provide a substantially coherent evanescent wave illumination of the object  1220 , the illumination  1210  characterized by a wavelength λ and a radius of curvature and disposed at one of a plurality of incident wave vectors from about 2π/λ to about 2πn sub /λ with respect to a surface normal of the substrate and at a plurality of azimuth angles spanning from about 0 to about 2π, wherein the plurality of incident wave vectors correspond to angles beyond a total internal reflection angle θ c  of the substrate. The apparatus  1200  can also include a second optical system  1230  disposed to collect portions of the illumination  1208  scattered from the object plane  1222 , the second optical system  1230  having an optical axis  1236  disposed at one of a plurality of center wave vectors from about 0 to about 2π/λ (or angles from about 0 to about π) with respect to the substrate  1225  surface normal and at the azimuth angle corresponding to the illumination of the first optical system, wherein the second optical system  1230  is characterized by a numerical aperture (NA).  FIG. 12B  shows arrangement with tilted optical axis  1236 ′. The apparatus  1200  can also include a third optical system disposed in an optical path of the first optical system to provide interferometric reintroduction of a portion of the coherent illumination (reference beam) into the second optical system  1230  or  1230 ′, wherein each of an amplitude, a phase, a radius of curvature and an angle of incidence of the reference is adjusted such that a correct reference wave is present at the image plane of the second optical system. The apparatus  1200  can further include an electronic image device disposed at an image plane of the second optical system that responds linearly to the local optical intensity and transfers the local optical intensity map across the image plane (a sub-image) to a signal processor device in electronic form. The apparatus  1200  can also include a device for adjusting the first, the second, and the third optical systems to collect sub-images for different pairs of the pluralities of incident (first optical system) and collection center (second optical system) wave vectors so as to sequentially obtain a plurality of sub-images corresponding to a plurality of regions of spatial frequency space and an electronic device to sequentially receive the electronic form of the sub-images and manipulate the sub-images to correct for distortions and alterations introduced by the optical configuration, store, and combine the plurality of sub-images corresponding to the plurality of regions of spatial frequency space to create a composite image corresponding to a synthetic aperture that is larger than the physical aperture of the collection lens  1230  or  1230 ′. 
     In some embodiments, the third optical system can further include a first beamsplitter disposed in the optical path of the first optical system before the object to collect a portion of the coherent illumination and one or more optics disposed between the first optical system and the second optical system  1230  are prisms  1262  within first optical system used to inject into substrate at angles beyond total internal reflection to interferometrically reintroduce the portion of the coherent illumination as a reference beam into the second optical system  1230  in a position after the exit aperture of a collection (objective) lens, wherein the reintroduction is at one of a position corresponding to a position a zero-order beam would have had if it had been transmitted through a higher NA lens of the second optical system  1230  or an aliased position to reduce pixel requirements of the electronic image device, wherein the signal processor is adjusted to compensate for this spatial frequency aliasing (the same concept as the local oscillator frequency introduced earlier). In other embodiments, the third optical system of the apparatus  1200  can include one of the configurations shown in  FIGS. 5A-5E . 
     In certain embodiments apparatus  1200  for microscopy with an IIM arrangement with illumination by evanescent waves extending from a substrate can also include at least one known reference object to cover a small part of the image field. 
     According to various embodiments, there is a method for microscopy by evanescent illumination through a substrate. The method can include providing an object  1220  disposed on a surface of a planar substrate  1225  characterized by a homogeneous refractive index (n sub ) and a surface normal  1226  and providing a first optical system disposed to provide an evanescent wave illumination of the object plane  1222  by providing a substantially coherent illumination of the object plane  1222 , the illumination characterized by a wavelength λ and a radius of curvature and disposed at one of a plurality of incident wave vectors from about 2π/λ to about 2πn sub /λ with respect to a surface normal of the substrate and at a multiplicity of azimuth angles spanning 0 to 2π, wherein the plurality of incident wave vectors correspond to angles beyond a total internal reflection angle θ c  of the substrate. The method can further include providing a second optical system  1230  having an optical axis  1236  disposed at one of a plurality of center wave vectors from about 0 to about 2π/λ with respect to the surface normal, wherein the second optical system  1230  is characterized by a numerical aperture (NA). The method can also include providing a third optical system disposed in an optical path of the first optical system to provide interferometric reintroduction of a portion of the coherent plane wave illumination (reference beam) into the second optical system  1230 , wherein the amplitude, phase, and position of the reintroduced illumination wave in the image plane of the second optical system  1230  can be adjusted. The method can further include recording a sub-image of the object  1220  at an object plane  1222  using an electronic image device, wherein the sub-image is formed as a result of interference of the scattering from the coherent plane wave illumination of the object and the reference beam; adjusting the first, the second, and the third optical systems to sequentially collect a plurality of sub-images corresponding to a plurality of regions of spatial frequency space; manipulating each of the plurality of sub-images using a signal processor to correct for distortions and alterations introduced by the optical configuration; and combining the plurality of sub-images into a composite image to provide a substantially faithful image of the object. In various embodiments, the method can further include one or more processes of subtraction of dark field images, subtraction of background images, shifting of spatial frequencies in accordance with the optical configuration, and elimination of one or more overlapping coverages of the frequency space wherein the elimination operations can be performed either in the optical systems or in the signal processing. In some embodiments, the method can also include selection of the regions of spatial frequency space to provide a more or less faithful image of the object in the object plane. Neumann et al. in Optics Express, Vol. 16, No. 25, 2008 pp 20477-20483 describes an evanescent wave illumination for further extending the resolution limit of imaging interferometric microscopy to λ\2(n+1), the disclosure of which is incorporated herein by reference in its entirety. 
     In various embodiments, the step of providing an object  1220  disposed on a surface of a planar substrate  1225  can include providing a cladding layer surrounding the object  1220  and the object  1220  disposed over the substrate  1225 . The extent of excitation region due to evanescent wave illumination, normal to the interface is given by an exponential decay function with a 1/e length of λ/2π√{square root over (n sub   2  sin 2  θ−n clad   2 )}, where n sub  is the refractive index of the substrate and n clad  is the refractive index of the superstrate or cladding material surrounding the object  1220 . The spatial localization can provide benefit, for example in TIRF (total internal reflection fluorescence) the localization is much larger than can be achieved with a simple focus or even with confocal microscopy. In other cases, this decay length can be a restriction, for example, in lithography studies where there might be multiple layers of material (bottom AR coating and photoresist for example) and the structural variation between these layers is of interest. Hence, the addition of a cladding layer surrounding the object can allow some degree of tuning of the decay length, and thereby control the signal to noise ratio. 
       FIGS. 13A-13C  shows several exemplary techniques for part of the first optical system to provide illumination through the substrate  1325 .  FIG. 13A  shows coupling of incident beam  1310  through a side  1327  of the substrate  1325 , which can be polished at an angle different from normal to the object  1320 ; in other words the substrate  1325  can be a prism.  FIG. 13B  shows one or more gratings  1364  on a side of the substrate  1325  the same as that where the object  1320  can be located. Alternatively, the gratings  1364  can be placed on a side opposite to that of the object  1320 .  FIG. 13C  shows coupling of the incident beam  1310  using one or more prisms  1362 . 
       FIG. 14A  shows a Manhattan (x-, y-geometry) test pattern, scaled to different dimensions. The Fourier intensity transform of this pattern for a linewidth (critical dimension or CD) of 180 nm is shown in  FIG. 14B  and for a CD of 150 nm in  FIG. 13C . The circles in  FIGS. 14B and 14C  correspond to the bandpass limits of various microscopy configurations. The circle in the center of  FIG. 14B , with a radius of NA/λ=0.4/λ, corresponds to the Abbé-limit spatial frequency range captured with on-axis coherent illumination (NA ill =0). The inner set of shifted circles in  FIG. 14B  (only single sidebands are shown for clarity; the complex conjugate regions are covered as well) correspond to IIM with off-axis illumination beams at θ ill =53° in the x, y-directions that extend the frequency coverage to a radius 3NA/λ˜1.2/λ. Additional frequency space coverage (second pair of circles) is available using evanescent wave illumination extending the frequency space coverage to a radius of (n sub  sin θ ill +NA)/λ˜1.87/λ (with θ ill =76° without tilt of the microscope optical axis. The frequency space coverage along with the corresponding structure Fourier intensity plot for the structure with CD=150 nm is shown in  FIG. 14C . The third pair of off-axis sub-images in  FIG. 14C  correspond to the tilted optical axis. This frequency region is elliptical rather than circular, due to nonparaxial and conical diffraction effects associated with the off-axis optical system. 
       FIG. 15A  shows the experimental result for an object containing both 180- and 170-nm CD structures in a single large-field image using the apparatus of  FIG. 12A  (two pairs of offset illumination, one at 53° in air and one at 76° in the substrate and collection with the optical axis along the substrate&#39;s surface normal as shown in  FIG. 12A . The 180-nm CD object is within the bandwidth capabilities of this optical system while the 170-nm CD object has significant spatial frequencies that extend beyond the optical system bandwidth and so is not fully resolved. The five nested “ell” shapes are distinguishable for the 180-nm CD, but not for the 170-nm CD. The positions of the two objects are correctly restored by the image restoration procedure as is evident from the good positional overlap between the experimental and theoretical cross-cuts in  FIG. 15B . 
       FIG. 16A  shows reconstructed high frequency image of a 150 nm structure using evanescent illumination and a tilted optical system, shown in  FIG. 12B , with the highest spatial frequencies collected with the optical axis tilted with respect to the substrate&#39;s surface normal.  FIG. 16B  shows high frequency image simulation of a 150 nm structure using evanescent illumination and a tilted optical system, shown in  FIG. 12B .  FIG. 16C  shows experimental and simulation cross-cuts of images shown in  FIGS. 16A and 16B .  FIG. 16D  shows reconstructed composite image of a 150 nm structure using evanescent illumination and a tilted optical system, shown in  FIG. 12B .  FIG. 16E  shows composite image simulation of a 150 nm structure using evanescent illumination and a tilted optical system, shown in  FIG. 12B .  FIG. 16F  shows experimental and simulation cross-cuts of images shown in  FIGS. 16D and 16E . 
     Evanescent illumination can be combined with structural illumination eliminating the need for access to the back focal plane. This moves the interferometer to the front of the objective lens and makes IIM readily adaptable to existing microscopes. Structural illumination is roughly equivalent to recording the spectral information at an intermediate frequency; additional computation is required to reset the frequencies. But this frequency shifting can reduce the camera pixel size and count requirements. Evanescent wave illumination can be used to extend the resolution of IIM to λ/2(n+1). Furthermore, IIM provides an important advantage over conventional immersion microscopy techniques. Since only a relatively small region of frequency space (˜NA/λ) is recorded in each sub-image, the aberration requirements on the objective lens are dramatically reduced. Hence, a simple set of prisms or gratings can be used to extract, and conventional air-based lenses to capture, the information. As is always the case, there is a trade-off between the number of sub-images and the NA of the objective lens. 
       FIG. 17  shows the possible increase of NA eff , drawn for a 0.4 NA system. As the frequency coverage is extended, the use of higher NA lenses can reduce the number of sub-images required for a more complete coverage of frequency space. Of course the required coverage is dependent on the pattern, and there are some applications, for example in metrology for integrated circuits, where coverage of a subset of the full frequency space is appropriate, where the range of spatial frequencies in the object are limited by lithographic consideration. 
     There are diffracted beams corresponding to even larger spatial frequencies (smaller features) scattered back into the substrate at angles larger than the critical angle. For planar substrate, these beams are totally internally reflected and are not accessible.  FIG. 18  shows another exemplary IIM optical arrangement for an apparatus  1800  for microscopy that provides access to the higher spatial frequency terms and thereby provides higher resolution, according to various embodiments of the present teachings. The apparatus  1800  can include an object plane  1822  on which can be disposed a first surface of a planar substrate  1825 , wherein the substrate  1825  is characterized by a homogeneous refractive index (n sub ) and a surface normal  1826 . The apparatus  1800  can also include a first optical system disposed to provide a substantially coherent illumination of the object plane, the illumination characterized by a wavelength λ, and a radius of curvature and disposed at one of a plurality of incident wave vectors from about 0 to about 2πn sub /λ with respect to a surface normal  1826  of the substrate  1825  and at a plurality of azimuth angles spanning from about 0 to about 2π. The apparatus  1800  can further include at least one grating  1864  on the side of the substrate  1825  opposite the object plane  1822 , wherein each grating  1864  is characterized by a period, a depth, a grating profile, a position, and an extent to further transform the scattered waves in the substrate reflected from the illumination by the object into propagating waves in the medium below the substrate. In some embodiments, the medium below the substrate  1825  can be air. In other embodiments, the medium can be a vacuum. However, the medium can include any other suitable material. The apparatus  1800  can further include a second optical system  1830  having an optical axis  1836  disposed at one of a plurality of center wave vectors from about 0 to about 2π/λ (or angles from about 0 to about π) with respect to the surface normal  1826 , wherein the second optical system  1830  can include one or more gratings  1864  on the second side of the substrate  1825  and is characterized by a numerical aperture (NA). The apparatus  1800  can also include a third optical system disposed in an optical path of the first optical system to provide interferometric reintroduction of a portion of the coherent illumination (reference beam) into the second optical system  1830 , wherein each of an amplitude, a phase, a radius of curvature, path length, and an angle of incidence of the reference can be adjusted such that a correct reference wave is present at the image plane of the second optical system. The apparatus  1800  can further include an electronic image device disposed at an image plane of the second optical system  1830  that responds linearly to the local optical intensity and transfers the local optical intensity map across the image plane (a sub-image) to a signal processor device in electronic form, a signal processor that receives the electronic form of the sub-image and manipulates the sub-image to correct for distortions and alteration introduced by the optical configuration, and to collect, store and combine a plurality of sub-images corresponding to a plurality of regions of spatial frequency space to create a composite image, wherein the plurality of sub-images are formed as a result of adjustments to the first, the second, and the third optical systems. In various embodiments, the third optical system of the apparatus  1800  can include one of the third optical system configurations shown in  FIGS. 5A-5E . 
     In various embodiments, the grating  1864  profile can have an impact on the extraction efficiency. In some embodiments, the grating  1864  can have a sinusoidal profile. A sinusoidal grating has components in its Fourier transform only at ±1/d. In other embodiments, the grating  1864  can have a rectangular profile. A rectangular grating has many more Fourier components that can lead to coupling of additional scattered image plane waves across the interface. For equal line: space grating, the second order Fourier coefficient (@±2/d) vanishes, although for sufficiently deep gratings, comparable to the wavelength, additional coupling terms can arise. The third order terms (at ±3/d) are always present for rectangular grating profiles. This can give rise to multiple coupling orders which can lead to artifacts in the sub-images. In some arrangements, this is not an issue because of the spatial separation of the scattered spatial frequency information at the bottom of the substrate (as can be seen in  FIG. 18 ). In this case, the bottom substrate plane is separated from the object plane and the different spatial frequency components, propagating at different angles, have separated to some extent by the time they reach this plane. If the thickness of the substrate  1825  is significantly larger than the field of view (illuminated aperture at the image plane), this separation can be large enough to avoid issues associated with higher order coupling at the bottom surface extraction grating. Thus, there is engineering trade-off in choosing the thickness of the substrate  1825 , the separation is better if it is thicker, but the phase distortions are increased. 
     Alternative collection schemes can include using one or more prisms  1974 , as shown in  FIG. 19 . In some embodiments, the prism  1974  can be fabricated as part of the substrate  1925 . In other embodiments, index matching fluid  1972  can be used. 
     In certain embodiments apparatus  1800  for microscopy can also include at least one known reference object to cover a small part of the image field. 
     According to various embodiments, there is a method for microscopy by illumination through a substrate. The method can include providing an object  1820  disposed over a first side of a planar substrate  1825 , the substrate characterized by a homogeneous refractive index (n sub ) and a surface normal  1826  such that the object  1820  is separated from the substrate  1825  by a distance of no more than about ≦λ. The method can also include providing at least one grating  1864  on the side of the substrate  1825  opposite the object plane  1822 , each grating  1864  characterized by a position, a period, a depth, and a grating profile, wherein each of the gratings  1864  can further scatter reflected waves resulting from the coherent illumination of the object into propagating waves in the medium below the substrate. The method can further include providing a first optical system disposed to provide a substantially coherent illumination of the object plane, the illumination characterized by a wavelength λ and a radius of curvature and disposed at one of a plurality of incident wave vectors from about 0 to about 2πn sub /λ with respect to a surface normal of the substrate and at a plurality of azimuth angles spanning from about 0 to about 2π. The method can also include providing a second optical system  1830  having an optical axis  1836  disposed at one of a plurality of center wave vectors from about 0 to about 2π/λ with respect to the surface normal  1826 , wherein the second optical system  1830  includes at least one grating  1864  on the second side of the substrate  1825  and is characterized by a numerical aperture (NA). The method can further include providing a third optical system disposed in an optical path of the first optical system to provide interferometric reintroduction of a portion of the coherent illumination (reference beam) into the second optical system  1830 , wherein each of an amplitude, a phase, a radius of curvature and an angle of incidence of the reference is adjusted as required such that a corrected reference wave is present at the image plane of the second optical system  1830 . The method can also include providing an electronic image device disposed at an image plane of the second optical system  1830  that responds linearly to the local optical intensity and transfers the local optical intensity map across the image plane (a sub-image) to a signal processor device in electronic form, providing a signal processor that receives the electronic form of the sub-image, manipulating each of the plurality of sub-images using the signal processor to correct for distortions and alterations introduced by the optical configuration, and combining the plurality of sub-images into a composite image to provide a substantially faithful image of the object. In various embodiments, the method can further include one or more processes of subtraction of dark field images, subtraction of background images, shifting of spatial frequencies in accordance with the optical configuration, and elimination of one or more overlapping coverages of the frequency space wherein the elimination operations can be performed either in the optical systems or in the signal processing. In some embodiments, the method can also include selecting regions of spatial frequency space to provide a more or less faithful image of the object in the object plane. 
     For various IIM configurations shown in  FIGS. 3 ,  12 , and  18 , the coherence length&gt;&gt;sample (object) dimensions. The He—Ne laser has a long coherence length of many cm, and this makes the experimental arrangement simpler, as it is not necessary to critically match the interferometer lengths between the objective arm and the zero-order reinjection arm. However, it does increase spurious speckle effects associated with stray light and multiple reflections from various optical surfaces in the set-up. These effects can be mitigated by choosing a source with sufficient coherence for the IIM measurements, but insufficient coherence for Fabry-Perot effects, e.g. between the front and back sides of the substrate or between the substrate and the objective entrance surface. Since, these dimensions are very different, μm scale for the sample to several mm for the substrate thickness and substrate-objective distance, it is possible to minimize unrelated Fabry-Perot effects while retaining all of the resolution of IIM. 
     Tiling of Frequency Space 
     In general, the spatial frequency location of the information corresponding to a specific angle of illumination (including illumination through the substrate) and angle of collection (θ) corresponds to 
     
       
         
           
             
               
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     In the above equation, (n) in the first term is adjusted as appropriate, for example, for illumination in air, n=1 while for illumination (evanescent) through a substrate, n=n sub =1.5 for glass. In keeping with the notation established above, θ scattered  is the angle in the substrate and so the factor n sub  is appropriate; a grating can be used to shift the spatial frequencies into the air propagation bandpass as necessary. 
     Both angles as well as the pitch of any gratings can provide some flexibility in the tiling of frequency space, i.e. in choosing the regions of frequency images into a complete image. The maximum spatial frequency, k max =2πf max =2π(2n sub /λ) is obtained when both angles are close to 90°. Since we can resolve a half pitch, this leads to an Abbe resolution limit of λ/4n sub . The optimum strategy is pattern dependent, for example, for Manhattan geometry structures with edges confined to a rectangular grid, often found in integrated circuits, it is important to capture the frequency information along the axes and of less consequence to capture the information away from the axes where the Fourier transform of the pattern has lower spectral intensity. In the examples shown in  FIGS. 19A and 19B , only one principal axis is considered, but the generalization to complete coverage is straightforward. 
       FIGS. 20A and 20B  show two alternate embodiments for providing coverage from f x =0 to f x =2n sub /λ with a fixed objective NA. The frequency space coverages shown in  FIGS. 20A and 20B  are designed for a maximum spatial frequency of 3/λ (2n sub /λ), and an objective of NA of 0.65. In  FIG. 20A , a minimum number of sub-images are used. The central small circle corresponds to conventional, normal incidence coherent illumination with the radius of the circle being about 0.65/λ. The next sub-image is taken with off-axis illumination through the substrate at an angle of about 53°; this corresponds to effective NA ill  of about 1.2. The scattered light can be detected either from the top (through air) or through the substrate, the collection geometry can be similar to that shown in  FIG. 18 , except for the illumination direction. A zero-order (interferometric reference) beam can be used in IIM to provide access to the essential phase information as well as to allow unambiguous assignment of the directly measured spatial frequencies. The square law, intensity detection process restores both the complex conjugate frequencies within the symmetrically located dotted circle in the figure. A third sub-image can be taken with grazing incidence illumination through the substrate, and with higher scattered spatial frequencies with the use of a grating of period λ/0.8 for extraction as in  FIG. 18 . Again, the complex conjugate spatial frequencies are restored by the square-law detection process. For a Manhattan geometry object, a similar set of sub-images in the orthogonal direction can be used, for a total of five sub-images; arbitrary structures may require additional sub-images to fill all of frequency space. 
       FIG. 20B  shows a second tiling embodiment, using four sub-images, but provides more robust coverage of frequency space (fewer missed spatial frequencies in the regions where the circles abut). The central circle is the same as in the previous example, illumination at normal incidence and collection with a conventional 0.65 NA optical system. The second innermost set of circles corresponds to illumination at grazing incidence in air (NA ill ˜1). The next innermost set corresponds to the same illumination condition, but to collection through a glass substrate. The final outermost set of circles corresponds to illumination at grazing incidence through the substrate and collection with the same grating to allow high spatial frequencies (collection of light scatted at angles beyond the critical angle in the glass). The disclosed exemplary embodiments provide an example of the flexibility offered by the IIM process. The choice of tiling strategy will depend on the object to be imaged. In general, it is best not to put a collection boundary in a region of high spectral intensity to minimize Gibbs effect oscillations of the observed sub-image structure. In addition, the strength of scattered spatial frequency components in the regions between the circles will be a factor in selecting an IIM tiling strategy. 
     It should be noted that the tiling with circular regions is not a requirement, but is convenient as a result of the symmetry of optical lenses. In some cases, a square aperture, which can be provided either optically or electronically during the sub-image manipulations, can prove advantageous. In particular, a square aperture can be configured to provide more robust coverage at the boundaries between sub-images (e.g. two squares can match along the line, while two circles can only touch at a point). The tilings in  FIG. 19B  show some overlap regions. Several strategies are available for dealing with the multiple counting in frequency space that these overlaps imply. The simplest is just to remove the double counting in the computation of the sub-image combination. Alternatively, a graded transfer function can be applied in the region of the overlap to minimize artifacts from imperfect cancellation of Gibbs effect oscillations in the two sub-images. The simplest approach is to calculate the Fourier transform of the sub-image, apply appropriate filters and inverse transform back to real space. The apparatus of image signal processing is very rich, and many of its techniques can be applied to this image reconstruction problem. 
     For arbitrary images, where a-priori information on likely orientations and spatial frequency content is not available, for example biological specimens, additional sub-images can be used in order to get a more complete coverage of spatial frequency space. An example of covering most of spatial frequency space is given in  FIG. 21 . This consists of 13 sub-images: the two off-axis sub-images shown in the top of  FIG. 20A  are repeated with rotation angles of 45°, 90° and 135° (there is no need to repeat the low-frequency sub-image) for a total of 9 sub-images; additional high frequency sub-images at rotation angles of 22.5°, 67.5°, 112.5°, and 157.5°, for a total of 13 sub-images, complete the coverage except for small regions near the outer periphery of frequency space. It should be noted that there are only three optical configurations; on-axis illumination (low frequency), middle frequency, and high frequency, the remaining sub-images are obtained by a simple sample rotation. Furthermore, provision can be made for illumination through the substrate for the middle and high frequency coverage as the sample is rotated. 
     The number of sub-images can be reduced by increasing the objective NA. As can be seen in  FIG. 22 , the number of sub-images for nearly full coverage is reduced to 5 for a NA=0.95, corresponding to a very high NA air-based objective. The specifics of the arrangement is that the low frequency sub-image is taken for normal incidence (NA ill =0); each of the offset sub-images is at NA ill +G/λ=2 which can be achieved with grazing incidence illumination through the substrate along with a grating with a period of λ/0.5. 
       FIG. 23  provides two similar  1 D tiling strategies for a silicon substrate (n sub =3.6 at 1064 nm), one (vertical) for a 0.65 NA and another (horizontal) for a 1.4 NA conventional immersion objective. As many as seven sub-images may be used to provide a complete coverage along just one axis for the 0.65 NA, whereas only three are sufficient for the large NA. The area of frequency space, and the required number of sub-images for nearly complete coverage, increases as n 2 . Scaling from  FIG. 20A  suggests that as many as (3.6/1.5) 2 ×13˜75 sub-images would be required for full coverage with the 0.64 NA objective. This suggests that there will be great advantage in knowing something about the image and its spectral content. One situation where this is clearly possible is in the inspection of silicon integrated circuits. The demands of manufacturable lithography at the nanoscale are forcing lithographers to restrict the range of patterns allowed in modern integrated circuits. This is often referred to as lithography “friendly” design, which in general is forcing the patterns closer to periodic grating patterns. In turn, a lithography “friendly” circuit is a microscopy “friendly” circuit with a limited range of spatial frequencies, hence complete coverage of spatial frequency space is not required to reconstruct an image. Immersion lenses are not available at an NA corresponding to the refractive index of silicon (3.6). An available immersion lens designed for more modest NAs of ˜1.4 can be used with the addition of gratings to couple the higher spatial frequency light out of the substrate. An issue with the very high NA immersion lens is that these lenses typically have a very short working distance, which in turn will require a very thin substrate, or a specially designed objective. 
     Imaging interferometric microscopy techniques as described above are sensitive to the optical refractive index variation of the object materials and does not contain any material specific information. Imaging interferometric microscopy can be applied to get material and chemical information using coherent anti-Stokes Raman scattering (CARS) spectroscopic microscopy. An apparatus for coherent anti-Stokes Raman (CARS) microscopy can include any suitable optical arrangement as shown in  FIGS. 1 ,  3 ,  5 A- 5 E,  12 A,  12 B  13 A- 13 C,  18 , and  19 . In particular, the apparatus for CARS microscopy can include an object plane  122 ,  222 ,  1222 ,  1822  on which can be disposed a first surface of a planar substrate  125 ,  225 ,  1225 ,  1825 , wherein the substrate  125 ,  225 ,  1225 ,  1825  can be characterized by a homogeneous refractive index (n sub ) and a surface normal  226 ,  1226 ,  1826 . The apparatus for CARS microscopy can also include a first optical system disposed to provide a illumination of the object plane  122 ,  222 ,  1222 ,  1822 , the illumination characterized by two substantially coincident coherent beams  110 ,  110 ′,  210 ,  210 ′, with wavelengths λ 1  and λ 2  and corresponding angular frequencies ω 1  and ω 2  with ω 1 &gt;ω 2 , a radius of curvature, and disposed at one of a plurality of incident wave vectors from about 0 to about 2πn sub /λ 1  with respect to a surface normal of the substrate  125 ,  225 ,  1225 ,  1825  and at a multiplicity of azimuth angles spanning 0 to 2π. The apparatus for CARS microscopy can also include a second optical system (collection)  130 ,  230 ,  530 ,  1230 ,  1830  having an optical axis  136 ,  236 ,  536 ,  1236 ,  1836  disposed at one of a plurality of center wave vectors from about 0 to about 2πn sub /λ 1  with respect to the surface normal, wherein the second optical system  130 ,  230 ,  530 ,  1230 ,  1830  is characterized by a numerical aperture (NA) and is responsive primarily to optical signals at frequencies greater than ω 1 . The apparatus for CARS microscopy can further include a third optical system disposed in an optical path of the first optical system to provide interferometric reintroduction of a reference illumination (reference beam) at a frequency of 2ω 1 −ω 2 , into the second optical system  130 ,  230 ,  530 ,  1230 ,  1830 , wherein each of an amplitude, a phase, a radius of curvature and an angle of incidence of the reference is adjusted as required such that a corrected reference wave is present at the image plane of the second optical system  130 ,  230 ,  530 ,  1230 ,  1830 . The apparatus for CARS microscopy can also include an electronic image device disposed at an image plane  124 ,  224  of the second optical system  130 ,  230 ,  530 ,  1230 ,  1830  that responds linearly to the local optical intensity and transfers the local optical intensity map across the image plane (a sub-image) to a signal processor device in electronic form, a signal processor that receives the electronic form of the sub-image and manipulates the sub-image to correct for distortions and alteration introduced by the optical configuration, and an electronic device to sequentially collect, store and combine a plurality of sub-images corresponding to a plurality of regions of spatial frequency space to create a composite image, wherein the plurality of sub-images are formed as a result of adjustments to the first, the second, and the third optical systems. 
     In various embodiments, the third optical system of the apparatus for CARS microscopy can include a first beamsplitter disposed in the optical path of the first optical system before the object plane  122 ,  222 ,  1222 ,  1822  to collect a portion of the coherent illumination and one or more optics disposed between the first optical system and the second optical system  130 ,  230 ,  530 ,  1230 ,  1830 , wherein the optics includes a nonresonant nonlinear material configured to generate the anti-Stokes four-wave mixing frequency 2ω 1 −ω 2  and exclude the fundamental frequencies (ω 1  and ω 2 ), and to interferometrically reintroduce the portion of the anti-Stokes coherent illumination as a reference beam into the second optical system  130 ,  230 ,  530 ,  1230 ,  1830  in a position after the exit aperture of a collection (objective) lens, wherein the reintroduction is at one of a position corresponding to a position a zero-order beam would have had if it had been transmitted through an appropriate higher NA lens of the second optical system  130 ,  230 ,  530 ,  1230 ,  1830  as shown in  FIG. 1  or an aliased position to reduce pixel requirements of the electronic image device, wherein the signal processor is adjusted to compensate for this spatial frequency aliasing. 
     In various embodiments, the third optical system of the apparatus for CARS microscopy can include one of the third optical system configurations shown in  FIGS. 5A-5E . In some embodiments, the apparatus for CARS microscopy can include a third optical system  500 E in a configuration shown in  FIG. 5E . The third optical system can include a first beamsplitter disposed in the optical path of the first optical system before the object plane  522  to collect a portion of the coherent illumination one or more transfer optics disposed between the first optical system and the second optical system  530 , wherein the optics includes a nonresonant nonlinear material  520  configured to generate the anti-Stokes four-wave mixing frequency 2ω 1 −ω 2  and exclude the fundamental frequencies (ω 1  and ω 2 ), and a second beamsplitter  570  disposed between the object plane  522  and a front aperture of a collection lens (objective) of the second optical system  530  to reintroduce the portion of the anti-Stokes coherent wave illumination as a reference beam  510 ′ into the second optical system  530  at an angle θ less than the entrance angular aperture (&lt;˜sin −1  NA) of the second optical system  530 . 
     In other embodiments, the apparatus for CARS microscopy can include a third optical system  500 D in a configuration shown in  FIG. 5D . The third optical system  500 D can further include a first beamsplitter disposed in the optical path of the first optical system to collect a portion of the coherent illumination, one or more transfer optics disposed between the first optical system and the second optical system, wherein the optics includes a nonresonant nonlinear material configured to generate the anti-Stokes four-wave mixing frequency 2ω 1 −ω 2  and exclude the fundamental frequencies (ω 1  and ω 2 ). The third optical system  500 D can also include at least one of a grating  584  or a grating on a waveguide disposed between the object plane  522  and a front aperture of the collection lens (objective) of the second optical system  530  to reintroduce the portion of the anti-Stokes coherent wave illumination as a reference beam  510 ′ into the second optical system  530  at an angle θ less than the entrance angular aperture (&lt;˜sin −1  NA) of the second optical system  530 . 
     In other embodiments, the apparatus for CARS microscopy can include a third optical system  500 A in a configuration shown in  FIG. 5A . The third optical system  500 A can further include a first beamsplitter disposed in the optical path of the first optical system to collect a portion of the coherent illumination, one or more transfer optics, wherein the one or more optics can include a nonresonant nonlinear material configured to generate the anti-Stokes four-wave mixing frequency 2ω 1 −ω 2  and exclude the fundamental frequencies (ω 1  and ω 2 ) and wherein at least one of the one or more optics is disposed to direct the portion of the anti-Stokes coherent plane wave illumination as a reference beam to illuminate the object at an angle θ corresponding to less than the entrance angular aperture (&lt;˜sin −1  NA) of the second optical system  530 . The third optical system  500 A can also include a dynamic (on/off) physical block  550  disposed in a back pupil plane of the second optical system  530  to alternately block and unblock a small portion of the pupil aperture corresponding to the position of the reference beam  510 ′ in the aperture. 
     In various embodiments, the apparatus for CARS microscopy can include a third optical system  500 C in a configuration shown in  FIG. 5C . The third optical system  500 C can further include a first beamsplitter disposed in the optical path of the first optical system to collect a portion of the coherent illumination, one or more transfer optics, wherein the one or more optics can include a nonresonant nonlinear material configured to generate the anti-Stokes four-wave mixing frequency 2ω 1 −ω 2  and exclude the fundamental frequencies (ω 1  and ω 2 ) and wherein at least one of the one or more optics is disposed to direct the portion of the anti-Stokes coherent plane wave illumination as a reference beam to illuminate the object at an angle θ corresponding to less than the entrance angular aperture (&lt;˜sin −1  NA) of the second optical system  530 . The third optical system  500 C can also include a guided-mode resonance filter (k-vector filter)  582  disposed between the object plane  522  and a collection lens of the second optical system  530  and an another device (not shown) to adjust the position, tilt and rotation of the guided-mode resonance filter  582  between positions, tilts and rotations in which it alternately transmits and blocks the portion of the reference beam transmitted through the object plane. 
     In certain embodiments, the apparatus for CARS microscopy can also include at least one known reference object to cover a small part of the image field. In some embodiments, the first, the second, and the third optical systems can be arranged in a transmission configuration. 
     In other embodiments, the first, the second, and the third optical systems can be arranged in a reflection configuration. In some embodiments, the plurality of incident wave vectors of the first optical system can include wave vectors less than about 2π/λ 1  wherein these wave vectors are accessed by illumination of the substrate at polar angles between 0 and π/2. In other embodiments, the plurality of incident wave vectors of the first optical system can include wave vectors between about 2π/λ 1  and about 2πn sub /λ 1 , wherein these wave vectors are accessed by evanescent wave illumination of the object through the substrate. Furthermore, the apparatus for CARS microscopy can use any of the arrangements shown in  FIGS. 13A-13C  for coupling light into the substrate for illumination through the substrate  125 ,  225 ,  1225 , and  1825 . 
     In some other embodiments, the plurality of center wave vectors of the second optical system  130 ,  230 ,  530 ,  1230 ,  1830  can include only center wave vectors less than about 2π/λ 1 , wherein these center wave vectors are accessed by an optical system above the object plane of the substrate  125 ,  225 ,  1225 ,  1825 . In certain embodiments, the plurality of center wave vectors of the second optical system  130 ,  230 ,  530 ,  1230 ,  1830  can include center wave vectors between 2π/λ 1  and 2πn sub /λ 1 , wherein the center wave vectors greater than 2π/λ 1  are accessed through the substrate  125 ,  225 ,  1225 ,  1825  and the second optical system  130 ,  230 ,  530 ,  1230 ,  1830  can include one or more gratings on the side of the planar substrate  125 ,  225 ,  1225 ,  1825  opposite the object plane  122 ,  222 ,  1222 ,  1822 , wherein each grating is characterized by a position, a pitch, and a grating profile. 
     According to various embodiments, there is a method for coherent anti-Stokes Raman (CARS) microscopy. The method for CARS microscopy can include providing an object  120 ,  220 ,  1220 ,  1820  disposed over a planar substrate  125 ,  225 ,  1225 ,  1825 , wherein the substrate  125 ,  225 ,  1225 ,  1825  is characterized by a homogeneous refractive index (n sub ) and a surface normal and providing a first optical system disposed to provide a illumination of the object plane  122 ,  222 ,  1222 ,  1822 , the illumination characterized by two substantially coincident coherent beams with wavelengths λ 1  and λ 2 □ and corresponding angular frequencies ω 1  and ω 2  with ω 1 &gt;ω 2 , a radius of curvature, and disposed at one of a plurality of incident wave vectors from about 0 to about 2πn sub /λ 1 □ with respect to a surface normal  126 ,  226 ,  1226 ,  1826  of the substrate  125 ,  225 ,  1225 ,  1825  and at a multiplicity of azimuth angles spanning 0 to 2π□. The method can also include providing a second optical system (collection)  130 ,  230 ,  1230 ,  1830  having an optical axis  136 ,  236 ,  1236 ,  1836  disposed at one of a plurality of center wave vectors from about 0 to about 2πnsub/λ 1 □ with respect to the surface normal  125 ,  225 ,  1225 ,  1825 , wherein the second optical system  130 ,  230 ,  1230 ,  1830  is characterized by a numerical aperture (NA) and is responsive primarily to optical signals at frequencies greater than ω 1  and providing a third optical system disposed in an optical path of the first optical system to provide interferometric reintroduction of a reference illumination (reference beam) at a frequency of 2ω 1 −ω 2 , into the second optical system  130 ,  230 ,  1230 ,  1830 , wherein each of an amplitude, a phase, a radius of curvature and an angle of incidence of the reference is adjusted as required such that a corrected reference wave is present at the image plane of the second optical system  130 ,  230 ,  1230 ,  1830 . The method can further include providing an electronic image device disposed at an image plane of the second optical system  130 ,  230 ,  1230 ,  1830  that responds linearly to the local optical intensity and transfers the local optical intensity map across the Image plane (a sub-image) to a signal processor device in electronic form, providing a signal processor that receives the electronic form of the sub-image, manipulating the sub-image using the signal processor to correct for distortions and alteration introduced by the optical configuration, providing an electronic device to sequentially collect, store and combine a plurality of sub-images corresponding to a plurality of regions of spatial frequency space to create a composite image, wherein the plurality of sub-images are formed as a result of adjustments to the first, the second, and the third optical systems, and combining the plurality of sub-images into a composite image to provide a substantially faithful image of the object  120 ,  220 ,  1220 ,  1820 . 
     According to various embodiments, the method can further include one or more processes of subtraction of dark field images, subtraction of background images, shifting of spatial frequencies in accordance with the optical configuration, and elimination of one or more overlapping coverages of the frequency space wherein the elimination operations can be performed either in the optical systems or in the signal processing. In some embodiments, the method can further include selecting regions of spatial frequency space to provide a more or less faithful image of the object  120 ,  220 ,  1220 ,  1820  in the object plane  122 ,  222 ,  1222 ,  1822 . 
     While the invention has been illustrated with respect to one or more implementations, alterations and/or modifications can be made to the illustrated examples without departing from the spirit and scope of the appended claims. In addition, while a particular feature of the invention may have been disclosed with respect to only one of several implementations, such feature may be combined with one or more other features of the other implementations as may be desired and advantageous for any given or particular function. Furthermore, to the extent that the terms “including”, “includes”, “having”, “has”, “with”, or variants thereof are used in either the detailed description and the claims, such terms are intended to be inclusive in a manner similar to the term “comprising.” As used herein, the term “one or more of” with respect to a listing of items such as, for example, A and B, means A alone, B alone, or A and B. 
     Other embodiments of the invention will be apparent to those skilled in the art from consideration of the specification and practice of the invention disclosed herein. It is intended that the specification and examples be considered as exemplary only, with a true scope and spirit of the invention being indicated by the following claims.