Abstract:
An interference contrast imaging system images phase objects. The system includes an illumination source, illumination optics, polarizing optics for splitting the illumination into orthogonal polarizations and for recombining the polarizations, objective optics that form an image at a detector, a wavefront coding element and a post processor for processing the image by removing a phase shift imparted by the wavefront coding element. The wavefront coding element has an aperture, is between the phase object and the detector, and provides an altered optical transfer function of the imaging system by imparting the phase shift to the illumination transmitted through the wavefront coding element. The altered optical transfer function is insensitive to an object distance between the phase object and the objective optics over a greater range of object distances than would be provided by an optical transfer function of a corresponding interference contrast imaging system without the wavefront coding element.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]     This patent application is a continuation of commonly-owned and copending U.S. patent application Ser. No. 10/355,761, filed on Jan. 31, 2003, which is a continuation of U.S. patent application Ser. No. 09/875,766, filed on Jun. 6, 2001, now abandoned, both of which applications are incorporated herein by reference.  
         [0002]     This patent application is also a continuation-in-part of commonly-owned and copending U.S. patent application Ser. No. 09/070,969,-filed on May 1, 1998; which is a continuation-in-part of U.S. patent application Ser. No. 08/823,894 filed Mar. 17, 1997, now U.S. Pat. No. 5,748,371; which is a continuation of U.S. patent application Ser. No. 08/384,257, filed Feb. 3, 1995, now abandoned, all of which are incorporated herein by reference.  
         [0003]     This patent application also relates to commonly-owned and copending U.S. patent application Ser. No. 09/766,325, filed Jan. 19, 2001, now U.S. Pat. No. 6,783,733, which is incorporated herein by reference. 
     
    
     BACKGROUND OF THE INVENTION  
       [0004]     1. Field Of The Invention  
         [0005]     This invention relates to apparatus and methods for using Wavefront Coding to improve contrast imaging of objects which are transparent, reflective or vary in thickness or index of refraction.  
         [0006]     2. Description Of The Prior Art  
         [0007]     Most imaging systems generate image contrast through variations in reflectance or absorption of the object being viewed. Objects that are transparent or reflective but have variations in index of refraction or thickness can be very difficult to image. These types of transparent or reflective objects can be considered “Phase Objects”. Various techniques have been developed to produce high contrast images from essentially transparent objects that have only variations in thickness or index of refraction. These techniques generally modify both the illumination optics and the imaging optics and are different modes of what can be called “Contrast Imaging”.  
         [0008]     There are a number of different Contrast Imaging techniques that have been developed over the years to image Phase Objects. These techniques can be grouped into three classes that are dependent on the type of modification made to the back focal plane of the imaging objective and the type of illumination method used. The simplest Contrast Imaging techniques modify the back focal plane of the imaging objective with an intensity or amplitude mask. Other techniques modify the back focal plane of the objective with phase masks. Still more techniques require the use of polarized illumination and polarization-sensitive beam splitters and shearing devices.  
         [0009]     Contrast Imaging techniques that require polarizers, beam splitters and beam shearing to image optical phase gradients, we call “Interference Contrast” techniques. These techniques include conventional Differential Interference Contrast (Smith, L. W., Microscopic interferometry, Research (London), 8:385-395, 1955), improvements using Nomarski prisms (Allen, R. D., David, G. B, and Nomarski, G, The Zeiss-Nomarski differential interference equipment for transmitted light microscopy, Z. Wiss. Mikrosk. 69:193-221, 1969), the Dyson interference microscope (Born and Wolf, Principals of Optics, Macmillan, 1964), the Jamin-Lebedeff interferometer microscopes as described by Spencer in 1982 (“Fundamentals of Light Microscopy”, Cambridge University Press, London), and Mach-Zehnder type interference microscopes (“Video Microscopy”, Inoue and Spring, Plenum Press, NY, 1997). Other related techniques include those that use reduced cost beam splitters and polarizers (U.S. Pat. No. 4,964,707), systems that employ contrast enhancement of the detected images (U.S. Pat. No. 5,572,359), systems that vary the microscope phase settings and combine a multiplicity of images (U.S. Pat. No. 5,969,855), and systems having variable amounts of beam shearing (U.S. Pat. No. 6,128,127).  
         [0010]      FIG. 1  (Prior Art) is a block diagram  100 , which shows generally how Interference Contrast Imaging techniques are implemented. This block diagram shows imaging of a Phase Object  110  through transmission, but those skilled in the art will appreciate that the elements could just as simply have been arranged to show imaging through reflection.  
         [0011]     Illumination source  102  and polarizer  104  act to form linearly polarized light. Beam splitter  106  divides the linearly polarized light into two linearly polarized beams that are orthogonally polarized. Such orthogonal beams can be laterally displaced or sheared relative to each other. Illumination optics  108  act to produce focussed light upon Phase Object  110 . A Phase Object is defined here as an object that is transparent or reflective but has variations in thickness and/or index of refraction, and thus can be difficult to image because the majority of the image contrast typically is derived from variations in the reflectance or absorbtion of the object.  
         [0012]     Objective lens  112  and tube lens  118  act to produce an image upon detector  120 . Beam splitter  114  acts to remove the lateral shear between the two orthogonally polarized beams formed by beam splitter  106 . Beam splitter  114  is also generally adjustable. By adjusting this beam splitter a phase difference between the two orthogonal beams can be realized. Analyzer  116  acts to combine the orthogonal beams by converting them to the same linear polarization. Detector  122  can be film, a CCD detector array, a CMOS detector, etc. Traditional imaging, such as bright field imaging, would result if polarizer and analyzer  104  and  116  and beam splitters  106  and  114  were not used.  
         [0013]      FIG. 2  (Prior Art) shows a description of the ray path and polarizations through the length of the Interference Contrast imaging system of  FIG. 1 . The lower diagram of  FIG. 2  describes the ray path while the upper diagram describes the polarizations. The illumination light is linearly polarized after polarizer  204 . This linear polarization is described as a vertical arrow in the upper diagram directly above polarizer  204 . At beam splitter  206  the single beam of light becomes two orthogonally polarized beams of light that are spatially displaced or sheared with respect to each other. This is indicated by the two paths (solid and dotted) in both diagrams. Notice that the two polarization states of the two paths in the top diagram are orthogonally rotated with respect to each other. Beam splitter  214  spatially combines the two polarizations with a possible phase offset or bias. This phase bias is given by the parameter Δ in the upper plot. By laterally adjusting the second beam splitter  214  the value of the phase bias Δ can be changed. A Nomarski type prism is described by the ray path diagram, although a Wollaston type prism could have been used as well. Analyzer  216  acts to convert the orthogonal component beams to linearly polarized light. The angle between the polarizer  204  and analyzer  216  can typically be varied in order to adjust the background intensity. Image plane  218  acts to display or record a time average intensity of the linearly polarized light, the sheared component possibly containing a phase shift. This image plane can be an optical viewing device or a digital detector such as CCD, CMOS, etc.  
         [0014]     The interactions of the polarizers, beam splitters, and Phase Objects of the Interference Contrast imaging systems have been studied in great detail. For additional background information see “Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging”, Cogswell and Sheppard, Journal of Microscopy, Vol 165, Pt 1, January 1992, pp 81-101.  
         [0015]     In order to understand the relationship between the object, image, and phase shift Δ consider an arbitrary spatially constant object that can be mathematically described as: 
 
 Obj=a  exp( j θ), where  j=√{square root over (−1)} 
 
 where “a” is the amplitude and θ is the object phase. If the two component beams of the system of  FIG. 2  have equal amplitude, and if the component beams are subtracted with relative phases ±Δ/2 then just after analyzer  216  the resulting image amplitude is given by: 
 
 amp=a  exp( j[θ−Δ/ 2])− a  exp( j[θ+Δ/ 2])=2  j a  exp( j θ)sin(Δ/2) 
 
         [0016]     The image intensity is the square of the image amplitude. The intensity of this signal is then given by: 
 
 int   o =4  a   2  sin(Δ/2) 2 . 
 
         [0017]     The image intensity is independent of the object phase θ. The phase difference or bias between the two orthogonal beams is given by Δ and is adjusted by lateral movement of the beam splitter, be it a Wollaston or a Nomarski type. If instead of a spatially constant object, consider an object whose phase varies by Δφ between two laterally sheared beams. This object phase variation is equivalent to a change in the value of the component beam phases of Δ. If the component beam phases Δ is equal to zero (no relative phase shift) then the resulting image intensity can be shown to have increases in intensity for both positive and negative variations of object phase. If the component beam bias is increased so that the total phase variation is always positive, the change in image intensity then increases monotonically throughout the range Δφ. The actual value of the change in image intensity with object phase change Δφ can be shown to be: 
 
 Int   1 =4  a   2  Δφ sin(Δ). 
 
         [0018]     In Interference Contrast imaging the phase bias Δ determines the relative strengths with which the phase and amplitude information of the object will be displayed in the image. If the object has amplitude variations these will be imaged according to into above. At a phase bias of zero (or multiple of 2 pi ) the image will contain a maximum of phase information but a minimum of amplitude information. At a phase bias of pi the opposite is true, with the image giving a maximum of amplitude information of the object and a minimum of phase information. For intermediate values of phase bias both phase and amplitude are imaged and the typical Interference Contrast bias relief image is produced, as is well known.  
         [0019]     Variation of the phase bias can be shown to affect the parameters of image contrast, linearity, and signal-to-noise ratio (SNR) as well. The ratio of contrast from phase and amplitude in Interference Contrast imaging can be shown to be given by: 
 
[contrast due to phase/contrast due to amplitude]=2 cot(Δ/2) 
 
         [0020]     The overall contrast in the Interference Contrast image is the ratio of the signal strength to the background and can be shown to be given by: 
 
overall contrast=2 Δφ cot(Δ/2). 
 
         [0021]     The linearity between the image intensity and phase gradients in the object can be described by: 
 
 L =[(1+sin(Δ)) (2/3) ]/[2 cos(Δ)]. 
 
         [0022]     The signal-to-noise ratio (SNR), ignoring all sources of noise except shot noise on the background, can be shown to be given by 
 
 SNR= 4  a  cos(Δ/2). 
 
         [0023]     In Interference Contrast imaging systems the condenser aperture can be opened to improve resolution, although in practice, to maintain contrast, the condenser aperture is usually not increased to full illumination. Imaging is typically then partially coherent. Description of the imaging characteristics for Interference Contrast imaging therefore needs to be expressed in terms of a partially coherent transfer function. The partially coherent transfer function (or transmission cross-coefficient), given as C(m,n;p,q), describes the strength of image contributions from pairs of spatial frequencies components m; p in the x direction and n; q in the y direction (Born and Wolf, Principals of Optics, Macmillan, 1975, p. 526). The intensity of the image in terms of the partially coherent transfer function image can be written as: 
 
 l ( x,y )=∫∫∫ T ( m,n ) T ( p,g )* C ( m,n;p,q )exp(2  pi j  [( m−p ) x +( n−q ) y ]) dm dn dp dq  
 
 where the limits of integration are +infinity to −infinity. The term T(m,n) is the spatial frequency content of the object amplitude transmittance t(x,y): 
 
 T ( m,n )=∫∫ t ( x,y )exp(2  pi j [mx+ny ]) dx dy  
 
 where again the limits of integration are +infinity to −infinity. ( )* denotes complex conjugate. When the condenser aperture is maximally opened and matched to the back aperture or exit pupil of the objective lens, the partially coherent transfer function reduces to (Intro. to Fourier Optics, Goodman, 1968, pg.120): 
 
 C ( m, n; p, q )=δ( m−n )δ( p−q )[ a  cos(ρ)−ρ sqrt {(1−ρ 2 )}]
 
 where ρ=sqrt(m 2 +p 2 ) and δ (x)=1 if x=0, δ (x)=0 otherwise. 
 
         [0024]     The effective transfer function for the Interference Contrast imaging system can be shown to be given as: 
 
 C ( m,n;p,q ) eff =2  C ( m,n;p,q ){cos[2  pi ( m−n )Λ]−cos(Δ)cos([2  pi ( m+n )Λ]−sin(Δ)sin [2  pi ( m+p )Λ]}
 
 where Λ is equal to the lateral shear of the beam splitters and C(m,n;p,q) is the partially coherent transfer function of the system without Interference Contrast modifications. 
 
         [0025]     Interference Contrast imaging is one of the most complex forms of imaging in terms of analysis and design. These systems are also widely used and studied. But, there is still a need to improve Interference Contrast Imaging of Phase Objects by increasing the depth of field for imaging thick objects, as well as for controlling focus-related aberrations in order to produce less expensive imaging systems than is currently possible.  
       SUMMARY OF THE INVENTION  
       [0026]     An object of the present invention is to improve Contrast Imaging of Phase Objects by increasing depth of field and controlling focus-related aberrations. This is accomplished by using Contrast Imaging apparatus and methods with Wavefront Coding aspheric optics and post processing to increase depth of field and reduce misfocus effects. The general Interference Contrast imaging system is modified with a special purpose optical element and image processing of the detected image to form the final image. Unlike the conventional Interference Contrast imaging system, the final Wavefront Coding Interference Contrast image is not directly available at the image plane. Post processing of the detected image is required. The Wavefront Coding optical element can be fabricated as a separate component, can be constructed as an integral component of the imaging objective, tube lens, beam splitter, polarizer or any combination of such.  
         [0027]     Apparatus for increasing depth of field and controlling focus related aberrations in an Interference Contrast Imaging system having an illumination source, optical elements for splitting light polarizations, and illumination optics placed before a Phase Object to be imaged, and elements for recombining light polarizations and objective optics after the Phase Object to form an image at a detector, includes an optical Wavefront Coding mask having an aperture and placed between the Phase Object and the detector, the coding mask being constructed and arranged to alter the optical transfer function of the Interference Contrast Imaging system in such a way that the altered optical transfer function is substantially insensitive to the distance between the Phase Object and the objective optics over a greater range of object distances than was provided by the unaltered optical transfer function, wherein the coding mask affects the alteration to the optical transfer function substantially by affecting the phase of light transmitted by the mask. The system further includes a post processing element for processing the image captured by the detector by reversing the alteration of the optical transfer function accomplished by the coding mask.  
         [0028]     The detector might be a charge coupled device (CCD).  
         [0029]     The phase of light transmitted by the coding mask is preferably relatively flat near the center of the aperture with increasing and decreasing phase near respective ends of the aperture.  
         [0030]     As an alternative, the phase of light transmitted by the coding mask could substantially follow a cubic function.  
         [0031]     In one embodiment, the phase of light transmitted by the coding mask substantially follows a function of the form: 
 
Phase ( x,y )=12 [ x   3   +y   3 ]
        where |x|≦1, |y|≦1.        
 
         [0033]     In another embodiment the phase of light transmitted by the coding mask substantially follows a rectangularly separable sum of powers function of the form: 
 
phase( x,y )=3 [ a   i  sign( x ) | x|   b     i     +c   i  sign( y ) | y|   d     i   ]
        where the sum is over the index i, 
 
sign( x )=−1 for  x&lt; 0, sign( x )=+1 for x≧0. 
       
 
         [0035]     In another embodiment, the phase of light transmitted by the coding mask substantially follows a non-separable function of the form: 
 
phase( r, θ)=3[ r    a     i    cos( b   i  θ+φ i )]
        where the sum is again over the index i.        
 
         [0037]     In another embodiment the phase of light transmitted by the coding mask substantially follows a function of the form: 
 
Phase profile ( x,y )=7[sign( x ) | x|   3 +sign( y ) | y|   3 +7[sign( x ) | x|   9.6 +sign( y ) | y|   9.6 ]
        where |x|≦1, |y|≦1.        
 
         [0039]     The coding mask further may be integrally formed with a lens element for focussing the light, or with the illumination optics.  
         [0040]     The coding mask could comprise an optical material having varying thickness, an optical material having varying index of refraction, spatial light modulators, or micro-mechanical mirrors.  
         [0041]     A method for increasing depth of field and controlling focus related aberrations in a conventional Interference Contrast Imaging system comprises the steps of modifying the wavefront of transmitted light between the Phase Object and the detector, the wavefront modification step selected to alter the optical transfer function of the Interference Contrast Imaging system in such a way that the altered optical transfer function is substantially insensitive to the distance between the Phase Object and the objective optics over a greater range of object distances than was provided by the unaltered optical transfer function, and post processing the image captured by the detector by reversing the alteration of the optical transfer function accomplished by the mask.  
         [0042]     A Wavefront Coding optical element can also be used on the illumination side of the system in order to extend the depth of field of the projected illumination due to the duality of projection and imaging. This projected illumination would be broader than without Wavefront Coding, but the optical density as a function of distance from the object would be less sensitive with Wavefront Coding than without. Without Wavefront Coding on the illumination side of the system, the object can technically be imaged clearly but is not illuminated sufficiently. See “Principal of Equivalence between Scanning and Conventional Optical Imaging Systems”, Dorian Kermisch, J. Opt. Soc. Am., Vol. 67, no. 10, pp. 1357-1360 (1977). 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0043]      FIG. 1  (prior art) shows a standard prior art Interference Contrast imaging system.  
         [0044]      FIG. 2  (prior art) shows ray paths and polarization states for the Interference Contrast imaging system of  FIG. 1 .  
         [0045]      FIG. 3  shows a Wavefront Coding Interference Contrast imaging system including Wavefront Coding optics and post processing in accordance with the present invention.  
         [0046]      FIG. 4  describes in detail the Object Modifying Function and Object Imaging Function of the Wavefront Coding Interference Contrast system.  
         [0047]      FIG. 5  shows the aperture transmittance function and the corresponding ambiguity function for the Object Imaging Function of the prior art system of  FIG. 1 .  
         [0048]      FIG. 6  shows the Wavefront Coded cubic phase function and the corresponding ambiguity function for the Object Imaging Function of  FIG. 3 .  
         [0049]      FIG. 7  shows another Wavefront Coded phase function and the corresponding ambiguity function for the Object Imaging Function of  FIG. 3 .  
         [0050]      FIG. 8  shows misfocus MTFs for the prior art Object Imaging Function of  FIG. 1  and the Object Imaging Functions for the Wavefront Coded Interference Contrast systems described in  FIGS. 3, 6  and  7 .  
         [0051]      FIG. 9  shows single plane of focus images of human cervical cells with darkly stained nuclei imaged with a 40X, NA=1.3 objective with a conventional Interference Contrast system and with a Wavefront Coded Interference Contrast imaging system similar to that of  FIG. 3 . 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0052]     Wavefront Coding can be used with conventional objectives, polarizers and beam splitters in Interference Contrast systems, as shown in  FIG. 3 , to achieve an increased depth of field in an optical and digital imaging system. This can be explained by considering the Object Modifying Functions of conventional Interference Contrast systems separately from the Object Imaging Functions, as shown in  FIG. 4 . By considering these two functions separately, modification of depth of field can be explained in terms of the Object Imaging Function. Extending the depth of field of the Object Imaging Functions of Interference Contrast systems is shown in  FIGS. 5-8 .  FIG. 9  shows real-world images of human cervical cells taken with a system having only Interference Contrast and a comparison to an image from a Wavefront Coding Interference Contrast system.  
         [0053]      FIG. 3  shows a Wavefront Coded Interference Contrast imaging system  300  including Wavefront Coding and post processing in accordance with the present invention. Similar reference numbers are used in  FIG. 3  as are used in  FIG. 1 , since the systems are very similar, except for the addition of Wavefront Coding element  324  and post processing  326 . The general Interference Contrast imaging system of  FIG. 1  is modified with a special purpose generalized aspheric optical element  324  and image processing  326  of the detected image to form the final image. Unlike the conventional Interference Contrast system, the final image in combined system  300  is not directly available at detector  322 . In fact, no sharp and clear image of any kind is available in system  300 , except after image processing  326 . Image processing  326  of the detected image is required to remove the spatial Wavefront Coding effects (other than the extended depth of field).  
         [0054]     Wavefront Coding optical element  324  can be fabricated as a separate component as shown in  FIG. 3 , or can be combined with objective lens  312 , tube lens  318 , beam splitter  314 , analyzer  316 , or any combination of these. Any material or configuration that can impart a range of spatial phase shifts to a wavefront can be used to construct Wavefront Coding element  324 . For example, optical glass or plastic of varying thickness and/or index of refraction can be used. Holograms and mirrors can also be used as the material for the Wavefront Coding element. In order to dynamically adjust the amount of depth of field, or to essentially change the Wavefront Coding element  324  for different objectives or desired depth of field, spatial light modulators or dynamically adjustable micro mirrors or similar can also be used.  
         [0055]     Wavefront Coding optical element  324  can also be used on the illumination side of system  300  in order to extend the depth of field of the projected illumination due to the duality of projection and imaging. This projected illumination would be broader than without Wavefront Coding, but the optical density as a function of distance from the object would be less sensitive with Wavefront Coding than without.  
         [0056]     The components that distinguish the Wavefront Coding Interference Contrast system of  FIG. 3  from a general or brightfield imaging system is polarizer  304 , beam splitter  306 , beam splitter  314 , analyzer  316 , and Wavefront Coding element  324  and image processing  326 . The polarizer, analyzer, and beam splitters essentially use phase to modify the imaging characteristics of the object  310 . The Wavefront Coding element  324  and image processing  326  are used to increase the depth of field or remove misfocus effects in images of the modified object as shown below. By grouping the components of system  300  by their function, the Wavefront Coding Interference Contrast imaging system can be understood.  
         [0057]     The locations of polarizer, analyzer, and beam splitters of  FIG. 3  have been chosen because of historical reasons. These are the traditional locations for these components in prior art systems relative to the illumination and imaging optics. The same relative locations are seen in  FIG. 1 . The beam splitter  314  and analyzer  316  can theoretically be moved relative to objective lens  312  without changing the imaging behavior of the system. See system  400 A of  FIG. 4 . Numbering conventions of  FIG. 4  are also similar to those of  FIGS. 1 and 3  due to the similar nature of the components. In system  400 A the beam splitter and analyzer have been moved before the objective lens but after the object. The wavefront after analyzer  416  is polarized as is the wavefront after analyzers  216  and  316  in  FIGS. 2 and 3  respectively. Since, ideally, lenses do not change the polarization, shear, or bias of the wavefront this new location is technically equivalent to that of  FIG. 3 . Consider the ray paths of  FIG. 2 . Notice that the ray paths between beam splitters  206  and  214  are parallel. Moving beam splitter  206  before objective lens  212  theoretically will not change the parallel nature-of the ray paths. Analyzer  216  can also move before objective lens  212  with no adverse affects. The component arrangement of system  400 A allows the “Object Modifying Functions” to be clearly distinguished from the Object Imaging Functions.  
         [0058]     In order to further characterize the Object Modifying Function of system  400 A consider system  400 B of  FIG. 4 . In this system a new phase and amplitude object  410 B replaces the original object  410 A of system  400 A. This new object is selected so that its three dimensional structure produces an identical wavefront from illumination source  402 , polarizer  404 , and illumination optics  408  as from object  410 A when combined with the polarizer, analyzer, and beam splitters of system  400 A. It is well known that a phase and amplitude object can be theoretically constructed so that any given linearly polarized wavefront can be reproduced from linearly polarized illumination. Although it is theoretically possible to produce such a new object  410 B, in practice it might be difficult. Since a new object  410 B can be substituted for the combination of original object  410 A, beam splitter  406 , beam splitter  414 , and analyzer  416 , it is clear that the polarizers and analyzers act to modify the imaging characteristics of the object. Notice that the right sides of systems  400 A and  400 B are identical. The right sides of these systems are the Object Imaging Function. The Object Imaging Function images the object that has had its imaging characteristics modified by the Object Modifying Function. With the Wavefront Coding optical element  424  and image processing  426  the Object Imaging Function can have a very large depth of field and be able to control focus-related aberrations.  
         [0059]     If the Object Imaging Function of system  400 B has a large depth of field, then the New Object of  410 B can be imaged over a large depth. Likewise, when the Object Imaging Function of system  400 A has a large depth of field, object  410 A (as modified by the Object Modifying Function) can be imaged with a large depth of field. Since system  400 B produces identical images to system  400 A, and system  400 A produces identical images to system  300 , this also means that system  300  will image object  310  with a large depth of field. This large depth of field is also independent of the object or Object Modifying Functions as shown in  FIG. 4 .  
         [0060]     The Object Imaging Function can be made to have a large depth of field by use of a generalized aspheric optical element and signal processing of the detected images. Ambiguity function representations can be used to succinctly describe this large depth of field. Only the magnitude of the ambiguity functions in this and following figures are shown. Ambiguity functions are, in general, complex functions. One-dimensional systems are given for simplicity. Those skilled in the art of linear systems and ambiguity function analysis can quickly make extensions to two-dimensional systems. An ambiguity function representation of the optical system is a powerful tool that allows modulation transfer functions (“MTFs”) to be inspected for all values of misfocus at the same time. Essentially, the ambiguity function representation of a given optical system is similar to a polar plot of the MTF as a function of misfocus. The in-focus MTF is described by the trace along the horizontal v=0 axis of the ambiguity function. An MTF with normalized misfocus value of Ψ=(2 π/λ)W 20 , where W 20  is the traditional misfocus aberration coefficient and λ is the illumination center wavelength, is described in the ambiguity function along the radial line with slope equal to (Ψ/pi). For more information on ambiguity function properties and their use in Wavefront Coding see “Extended Depth of Field Through Wavefront Coding”, E. R. Dowski and W. T. Cathey, Applied Optics, vol. 34, no 11, pp.1859-1866, April, 1995, and references contained therein.  
         [0061]      FIG. 5  gives an ambiguity function perspective on the Object Imaging Function of conventional Interference Contrast systems. The top plot of  FIG. 5  shows the aperture transmittance function of an ideal conventional Interference Contrast system such as that shown in  FIG. 1 . The bottom plot of  FIG. 5  shows the associated ambiguity function associated with the Object Imaging Function for the prior art system of  FIG. 1 .  
         [0062]     Over the normalized aperture (in normalized coordinates extending from −1 to +1) the conventional system has a transmittance of 1, i.e., 100%. The phase variation (not shown) is equal to zero over this range. The corresponding ambiguity function has concentrations of optical power (shown as dark shades) very close to the horizontal v=0 axis. From the relationship between the ambiguity function and misfocused MTFs, we see that the conventional Interference Contrast Systems has a small depth of field because slight changes in misfocus lead to MTFs (represented by radial lines with non-zero slope in the ambiguity function) that intersect regions of small power.  
         [0063]      FIG. 6  shows an example of a phase function for the Wavefront Coding optical element  324  and corresponding ambiguity function for an improved system of  FIG. 3 . This phase function is rectangularly separable and can be mathematically described in two dimensions as: 
 Phase ( x,y )=12 [ x   3   +y   3   ]|x|≦ 1, | y|≦ 1.  
         [0064]     Only one dimension of this phase function is shown in the upper plot of  FIG. 6 . Increasing the peak-to-valley phase height (as can be done by increasing the constant  12  above), results in increasing depth of field. The transmittance of this system (not shown) is unity (i.e., 100%) over the entire aperture, as in the top plot of  FIG. 5 .  
         [0065]     The ambiguity function shown in  FIG. 6  for this Wavefront Coded Interference Contrast system is seen to have optical power spread over a much larger region in the ambiguity domain than does the diffraction-limited system plotted in  FIG. 5 . Broader regions of optical power in the ambiguity function translate to larger depth of field or depth of focus since the ambiguity function is essentially a radial plot of misfocused MTFs with the angular dimension pertaining to misfocus. Thus, this Wavefront Coded Interference Contrast system has a larger depth of field than the conventional Interference Contrast system.  
         [0066]     There are an infinite number of different Wavefront Coding phase functions that can be used to extend the depth of field. Other more general rectangularly separable forms of the Wavefront Coding phase function are given by: 
 
phase( x,y )=3 [ a   i  sign( x ) | x|   b     i     +c   i  sign( y ) | y|   d     i   ]
        where the sum is over the index i, 
 
sign( x )=−1 for  x&lt; 0, sign( x )=+1 for  x≧ 0. 
       
 
         [0068]     Rectangularly separable forms of Wavefront Coding allow fast processing. Other forms of Wavefront Coding complex phases are non-separable, and the sum of rectangularly separable forms. One non-separable form is defined as: 
 
phase( r, θ)=3[ r    a     i    cos( b   i θ+φ i )]
        where the sum is again over the index i.        
 
         [0070]      FIG. 7  shows the Wavefront Coding phase function and the ambiguity function for a further improved system of  FIG. 3 . The top plot of  FIG. 7  shows the phase function from  FIG. 6  (curve  701 ) and a further improved phase function (curve  702 ). The aperture transmittance function is the same as shown in  FIG. 5 . The form of the new phase profile  702 , in radians, of this system is given by: 
 Phase profile ( x,y )=7[sign( x ) | x|   3 +sign( y ) |y| 3 +7[sign( x ) | x|   9.6 +sign( y ) | y|   9.6 ]        where |x|≦1, |y|≦1.          
         [0072]     The ambiguity function related to phase function  702  is shown in the bottom of  FIG. 7 . This ambiguity function is seen to have more optical power uniformly spread about the horizontal v=0 axis when compared to either the Wavefront Coding Interference Contrast system plotted in  FIG. 6  or the Conventional Interference Contrast system plotted in  FIG. 5 . Thus, the Wavefront Coded Interference Contrast system of  FIG. 7  will have a larger depth of field than the systems represented in FIGS.  6  or  5 .  
         [0073]      FIG. 8  shows MTFs as a function of misfocus for the prior art Interference Contrast system, and the Wavefront Coded Interference Contrast systems of  FIGS. 6 and 7 . The top plot of  FIG. 8  shows the MTFs of the conventional Interference Contrast imaging system of  FIG. 1  and  FIG. 5  and the MTFs of the Wavefront Coded Interference Contrast system of  FIG. 6 . The bottom plot shows the MTFs of the Interference Contrast imaging system of  FIGS. 1 and 5  (again) and the MTFs from the Wavefront Coding Interference Contrast imaging system of  FIG. 7 . These plots are the particular MTFs given in the respective ambiguity functions for the normalized misfocus values Ψ={0, 2, 4}. Notice that the MTFs for the conventional Interference Contrast system (top and bottom plots) vary appreciably with even this slight amount of misfocus. The image from the conventional system will thus change drastically due to misfocus effects for only small, misfocus values. This is expected from the narrow ambiguity function associated with the conventional system (shown in  FIG. 5 ).  
         [0074]     By comparison, the MTFs from the Wavefront Coded Interference Contrast imaging systems (top and bottom plots) show very little change with misfocus as predicted by the ambiguity functions associated with these systems (shown in  FIGS. 6 and 7 ). If the MTFs of the system do not change, the resulting MTFs (and hence also point spread functions, or “PSFs”) can be corrected over a large range of misfocus with a single post processing step  326 . A single post processing step is not possible with conventional systems, which change appreciably with misfocus since the MTFs and PSFs of the system change with misfocus to values that are unknown and often impossible in practice to calculate. The MTFs from the Wavefront Coded Interference Contrast system in the top plot are seen to have lower values for most spatial frequencies than the MTFs from the Wavefront Coded Interference Contrast system of the bottom plot. This is expected from the ambiguity functions of  FIGS. 6 and 7  respectively. The two-term phase function (curve  702 ) yields MTFs that not only have similarly small change with misfocus but also give a higher MTF than those associated with the simple cubic phase function (curve  701 ). This higher MTF results in a more compact PSF (not shown) as well as less signal-to-noise ratio penalties needed for the image processing  326 .  
         [0075]     In general, the Wavefront Coded objective mask phase function that yields the smallest MTF variation with misfocus and also the highest MTF is preferred in practice. There are an infinite number of different objective mask phase functions that are good candidates for control of the MTF. The characteristics that practical Wavefront Coding mask phase functions have can generally be described as being relatively flat near the center of the aperture with increasing and decreasing phase near the respective edges of the aperture. The central portion of the phase function controls the majority of the light rays that would not need modification if the objective were stopped down, for the depth of field extension required. For increasing amounts of depth of field, the size of the central phase region that can be flat decreases. Increasing the flatness of the central region of the rays leads to larger MTFs as seen in comparison to the phase functions and MTFs of  FIGS. 6, 7 , and  8 . The edge portion of the phase function controls the light rays that increase the light gathering and spatial resolution of the full aperture system but, without modification, cause the largest amount of misfocus effects in traditional systems. It is these edge rays that should be modified most by the objective mask phase function because they control the variation of the MTFs and PSFs with misfocus. The actual modification made to these edge rays should position them so that the sampled PSFs and MTFs are maximally insensitive to changes in misfocus.  
         [0076]     Notice that the MTFs from the Wavefront Coding Interference Contrast system of  FIG. 8  (upper and lower plots) essentially do not change with misfocus but also do not have the same shape as that of the in-focus MTF (Ψ=0) of the conventional Interference Contrast system. In the spatial domain, the Wavefront Coding Interference Contrast systems form images with a specialized blur where the blur is insensitive to the amount of misfocus. The Image Processing function  326  is used to remove-this blur. The Image Processing function can be designed so that after processing the MTFs and PSFs of the combined Wavefront Coding Interference Contrast system, over a range of misfocus, closely match that of the in-focus Interference Contrast system. The Image Processing function can also produce an effective MTF that has more or less contrast than the in-focus Interference Contrast system, depending on the needs of the particular application.  
         [0077]     In essence, the image processing function restores the Wavefront Coding Interference Contrast transfer functions to those expected from the conventional Interference Contrast system with no misfocus. Since all the Wavefront Coding MTFs are essentially identical, after image processing  326  all MTFs (and hence all PSFs) will be nearly identical for each value of misfocus.  
         [0078]     More specifically, the image processing function, say F, implements a transformation on the blurred Wavefront Coding Interference Contrast system, say H WFC , so that after processing the system has an ideal response H ideal . Typically the ideal response is chosen as the in-focus response from the general Interference Contrast system. If implemented as a linear filter, then F is (in the spatial frequency domain) equivalent to: 
 
 F ( w ) H   WFC ( W )= H   ideal ( w ) 
 
 where w denotes a spatial frequency variable. If the ideal response is fixed then changing the Wavefront Coding Interference Contrast system H WFC  changes the image processing function F. The use of a different Wavefront Coding phase function can cause a change in the image processing function. In practice, it is common to be able to measure slight changes in the Wavefront Coding Interference Contrast system as a function of misfocus. In this case the image processing F is chosen as a best fit between the measured data and the desired system after processing. 
 
         [0079]     There are many linear and non-linear prior art techniques for removing known and unknown blur in images. Computationally effective techniques include rectangularly separable or multi-rank linear filtering. Rectangularly separable linear filtering involves a two step process where the set of one-dimensional columns are filtered with a one-dimensional column filter and an intermediate image is formed. Filtering the set of one-dimensional rows of this intermediate image with a one-dimensional row filter produces the final image. Multi-rank filtering is essentially the parallel combination of more than one rectangularly separable filtering operation. A rank N digital filter kernel can be implemented with rectangularly separable filtering by using N rectangularly separable filters in parallel.  
         [0080]     The form of the processing (rectangularly separable, multi-rank, 2D kernel, etc.) is matched to that of the Wavefront Coding element. Rectangularly separable filtering requires a rectangularly separable Wavefront Coding element. The element described in  FIG. 6  is rectangularly separable.  
         [0081]      FIG. 9  contains real world images of human cervical cells made with a conventional Interference Contrast system and a Wavefront Coded Interference Contrast System. The image on the left of  FIG. 9  was made with a conventional 40X, NA=1.3 Interference Contrast system similar to that of  FIG. 1 . The image on the right of  FIG. 9  was made with a Wavefront Coding Interference Contrast system similar to that of  FIG. 3 . The Wavefront Coding Element  324  was a rectangularly separable cubic phase element. Rectangularly separable digital filtering was used for image processing  326 .  
         [0082]     Notice the phase shading visible in the conventional image. This phase shading results in a 3D-like appearance of the object. This is a characteristic of Interference Contrast imaging. Notice also that many parts of the Interference Contrast images are blurred due to misfocus effects. The bottom part of the left image, for example, is particularly badly blurred by misfocus. The Wavefront Coded Interference Contrast image is also seen to have similar phase shading and 3D-like appearance as the conventional image. The depth of field visible in the image is much larger in the Wavefront Coded image than in the conventional image. Many parts of the cells that could not be resolved in the conventional image are clearly visible in the Wavefront Coding image. Thus, the Wavefront Coding Interference Contrast image produces both the characteristic Interference Contrast phase object imaging characteristics and a large depth of field.  
         [0083]     As shown in  FIGS. 6 through 9 , the Wavefront Coding Interference Contrast imaging system removes the effects of misfocus on the final images. The Wavefront Coding Interference Contrast system will control the misfocus effects independent of the source of the misfocus. When increasing the depth of field, as shown in  FIG. 9 , the misfocus effects are produced from the object or parts of the object not being in the best focus position relative to the imaging optics. Misfocus effects can also be produced by non-ideal optics, temperature changes, mechanical positioning errors, and similar causes that lead to optical aberrations. Controlling misfocus effects besides those related to object positioning allows inexpensive systems to be produced that image with a high quality. For example, if the objective lens  312  of  FIG. 3  has a noticeable amount of chromatic aberration then misfocus effects will be produced as a function of illumination wavelength. The Wavefront Coding Interference Contrast system can control the chromatic aberration misfocus effects while also extending the depth of field. Other optical aberrations that can similarly be controlled include petzval curvature, astigmatism, spherical aberration, temperature related misfocus, and fabrication or alignment related misfocus. Many other aberrations in prior art systems may be improved in Wavefront Coding Interference Contrast systems