Patent Publication Number: US-9417610-B1

Title: Incoherent digital holographic adaptive optics

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a divisional of copending U.S. utility application entitled, “Incoherent Digital Holographic Adaptive Optics,” having Ser. No. 13/872,633, filed Apr. 29, 2013, which claims priority to U.S. Provisional Application Ser. No. 61/639,235, filed Apr. 27, 2012, which are entirely incorporated herein by reference. 
    
    
     BACKGROUND 
     Light from distant stars and galaxies travel through the near perfect vacuum of space for millions of years carrying the information about the distant parts of the universe until the last fraction of a second when it plunges through the earth&#39;s atmosphere before reaching our eyes or telescopes on the ground. Turbulence within the atmosphere causes degradation or aberration of images such that increasing the aperture of a telescope does not lead to improvement of resolution. Placing the telescope in space above the atmosphere or on top of mountains alleviates the problem but at considerable cost and constraint. Adaptive optics (AO) have been used over the past several decades to compensate for such aberration. When AO are used, the wavefront is sensed and adjusted by the optical system in real time to compensate for any distortion. Remarkable astronomical images have been obtained using ground-based telescopes with AO compensation of atmospheric turbulence, whose quality can even surpass space-based telescopes under favorable conditions. The principle of adaptive optics has also been applied to other imaging systems, most notably in ophthalmic imaging as well as in laser beam forming and remote sensing. 
     In a typical AO system, a guide star of sufficient brightness is used for measuring the aberration. A common method of wavefront sensing involves the use of a Shack-Hartmann sensor consisting of a lenslet array and a CCD camera underneath it. Distortions of the wavefront lead to shifting of lenslet focal spots proportional to the local slope of the wavefront. This shifting is used to compute the necessary displacement of a deformable mirror or other spatial light modulator. The wave-front sensing, wave-front modulation, and control subsystems form a closed loop to achieve a configuration that minimizes the aberration in the image of the full field. 
     While AO systems can greatly reduce distortion, they are not without their drawbacks. For example, AO systems are typically complex, high-precision systems that are both expensive and cumbersome. In addition, AO systems are also often slow to operate and prone to faults. It can therefore be appreciated that it would be desirable to have an alternative way to compensate for aberration. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present disclosure may be better understood with reference to the following figures. Matching reference numerals designate corresponding parts throughout the figures, which are not necessarily drawn to scale. 
         FIG. 1  is a schematic diagram of a first embodiment of a system for performing incoherent digital holographic adaptive optics (IDHAO). 
         FIG. 2  is a schematic diagram of a second embodiment of a system for performing IDHAO. 
         FIG. 3  illustrates a simulation of incoherent digital holography (IDH) and includes images of ( a ) an object field, ( b ) a raw inteferogram of the object field, ( c ) the amplitude of a complex hologram of the object field, ( d ) the phase of the complex hologram, ( e ) the amplitude of a reconstructed image, and ( f ) the phase of the reconstructed image. 
         FIG. 4  illustrates a simulation of adaptive optics using IDH and includes images of ( a ) an object field, ( b ) an assumed phase aberration, ( c ) a guide star inteferogram, ( d ) a full-field interferogram, ( e ) the amplitude of a complex hologram of the guide star, ( f ) the phase of the complex hologram of the guide star, ( g ) the amplitude of a complex hologram of the object field, ( h ) the phase of the complex hologram of the object field, ( i ) a direct image without the holographic process, ( j ) the amplitude of an IDH image without aberration compensation, ( k ) the phase of the IDH image without aberration compensation, ( l ) the amplitude of the complex hologram for the object field with aberration compensation, ( m ) the phase of the complex hologram for the object field with aberration compensation, ( n ) the amplitude of the IDH image with aberration compensation, ( o ) the phase of the IDH image with aberration compensation. 
         FIGS. 5( a ) and ( b )  together are a flow diagram of a method for generating aberration-compensated images using IDH. 
         FIG. 6  illustrates the resolution of reconstructed images versus focal length and includes images ( a ) without aberration and ( b ) with aberration for focal lengths of ( i ) 5000, ( ii ) 3000, ( iii ) 2000, and ( iv ) 1000 mm. 
         FIG. 7  illustrates IDHAO versus strength of aberration and includes images of. In each row are shown the phase aberration, the uncompensated image, the compensated image, and the direct image. The strength of the aberration is varied as ( a ) a ψ =−0.1, ( b ) a ψ =−0.2, ( c ) a ψ =−0.5, and ( d ) a ψ =−1.0. 
         FIG. 8  illustrates IDHAO versus various types of aberration: ( a ) ψ=(−0.5)×Z 1   +1 +(−0.5)×Z 2   0  ψ=(−0.5)×Z 1   +1 +(−0.5)×Z 2   0 , ( b ) ψ=(−0.5)×Z 1   +1 +(−0.5)×Z 3   +1 , ( c ) ψ=(−0.5)×Z 3   +3 +(−0.5)×Z 5   −5 , and ( d ) ψ=(−0.7)×Z 2   0 +(−0.5)×Z 5   −5 . 
         FIG. 9  illustrates IDHAO of an extended gray-scale object and includes images of ( a ) the assumed phase aberration, ( b ) the object, ( c ) the uncompensated image, and ( d ) the compensated image. 
         FIG. 10  illustrates the effect of a noisy hologram on IDHAO. In each row the phase of G ψ  and H ψ  and the amplitude of I ψ  and I ψ  are shown for ( a ) a ψ =0, a Φ =5, ( b ) a ψ =0, a Φ =10, ( c ) a ψ =0.2, a Φ =5, and ( d ) a ψ =0.2, a Φ =10. 
         FIG. 11  illustrates IDH without aberration and includes images of ( a ) a raw hologram, ( b ) a complex hologram, ( c ) a reconstructed image at z i =22000 mm, corresponding to z 1 =650 mm, and ( d ) a reconstructed image at z   i =14870 mm, corresponding to z   1 =440 mm. 
         FIG. 12  illustrates IDHAO with a phase aberrator placed just behind the objective lens and includes images of ( a ) a raw hologram, ( b ) a complex hologram, ( c ) a reconstructed image at z i , ( d ) a reconstructed image at z   i ;  e ) a guide star hologram of an LED, ( f ) a corrected hologram, ( g ) a corrected image at z i , and ( h ) a corrected image at z   i . 
         FIG. 13  illustrates IDHAO versus noise and includes images of ( a )-( c ) the object field with varying brightness obtained using ( i ) IDH without AO and ( ii )-( iv ) IDHAO with varying brightness of the guide star. 
         FIG. 14  illustrates IDH of extended objects and includes images of a resolution target (top row) and a knight chess piece (bottom row) illuminated with a miniature halogen lamp. In each case, a raw hologram and the amplitude and phase of the complex hologram are shown. 
         FIG. 15  illustrates IDHAO versus noise and includes images ( a )-( c ) of a resolution target with varying brightness obtained using ( i ) IDH without AO and ( ii )-( iv ) IDHAO with varying brightness of the guide star. 
         FIG. 16  illustrates three-dimensional focusing of IDHAO of an extended object and includes images of ( a ) an uncorrected image reconstructed at z   i =17500 mm, corresponding to z   1 =450 mm, ( b ) a reconstructed image at z i =22000 mm, corresponding to z 1 =650 mm, and ( c ) and ( d ) corresponding images with compensation. 
     
    
    
     DETAILED DESCRIPTION 
     As described above, it would be desirable to have a way to compensate for aberration other than through the use of conventional adaptive optics that rely on a deformable mirror or other spatial light modulator (SLM). Disclosed herein are systems and methods for generating aberration-compensated images using incoherent digital holography. In some embodiments, complex holograms are obtained for the full object field and for a guide star within the object filed by combining raw interferograms of the full field and the guide star, respectively. In some embodiments, the interferograms are obtained by performing a phase-shifting procedure using an interferometer having a linearly-displaceable planar mirror and a stationary curved mirror. Once the complex holograms are obtained for the full field and the guide star, mathematical correlation can be performed between the holograms to obtain an aberration-compensated image having reduced distortion. 
     In the following disclosure, various specific embodiments are described. It is to be understood that those embodiments are example implementations of the disclosed inventions and that alternative embodiments are possible. All such embodiments are intended to fall within the scope of this disclosure. 
     Described in the present disclosure is a new method of adaptive optics (AO) that can be used to measure and compensate aberrations based on incoherent digital holography with an interferometer. As described below, a hologram of a guide star contains sufficient information to compensate the effect of aberration on the full-field hologram. Although reconstruction from uncompensated full-field holograms yield distorted images comparable to direct nonholographic images formed through the same aberrating medium, the compensated holographic images have much of the aberrations removed. Provided in the disclosure that follows are descriptions of the theoretical, simulation, and experimental studies of the method, which is referred to herein as incoherent digital holographic adaptive optics (IDHAO). As will be apparent from this disclosure, the IDHAO process is robust and efficient under various ranges of parameters such as aberration types and strength. Furthermore, low image signals buried under noise can be extracted by IDHAO to yield high-quality images with good contrast and resolution, both for point-like sources and continuous extended objects illuminated with common incoherent light sources. 
     A theoretical description of IDHAO can be developed by considering the propagation of spherical waves from object source points through an interferometer, followed by integration of the interference intensities over the source points. A complex hologram is obtained by invoking a phase-shifting procedure, leading to an expression of the holographic image as a convolution of object intensity with a complex point spread function (PSF). Even in the presence an of an aberrating layer, under a specific condition, one obtains an holographic image as a convolution but with distorted PSF. A straightforward procedure is described below to compensate for the aberration by combining holograms of the full-field and a guide star. 
       FIG. 1  illustrates an example system  10  for performing IDHAO, which can also be used to simply perform incoherent digital holography (IDH). The system  10  is based on a variation of an interferometer in which the pixel-sharing SLM is replaced with a modified Michelson interferometer. As shown in  FIG. 1 , the system  10  includes a beam splitter BS that produces copies of a spherical wave emanating from each point source of an object field. In this example, the object field is a group of stars S that are imaged with a telescope (not shown). The copies of the spherical wave are provided to two mirrors, M A  and M B , which are both equidistant from the beam splitter BS. Both M A  and M B  are curved (concave) mirrors and therefore apply curvatures to the incident spherical waves. However, the curvatures of the two mirrors M A  and M B  are different from each other. In some embodiments, the mirror M A  has a focal length of approximately 1000 mm to infinity, while the mirror M B  has a radius of curvature of approximately 1000 to 1200 mm and, therefore, a focal length of approximately 500 to 600 mm. The position of the mirror M A  can be adjusted along the optical axis using a linear actuator  12 , such as a piezoelectric actuator, to which the mirror is mounted. The mirrors M A  and M B  reflect the incident waves to a camera  14  of the system  10 . In some embodiments, the camera  14  comprises a charge-coupled device (CCD). 
     Two copies of the same optical field are always coherent with each other. Therefore, when the reflected waves are superposed at the camera plane they produce a Fresnel zone-like interference pattern. Specifically, consider a point source of strength I o =|E o | 2  on the Σ o (x o , y o ) plane a distance z o  from the curved mirrors. First, consider IDH imaging without the aberrator Ψ in place. The field reflecting from M A  and arriving at the camera plane Σ c (x c , y c ) a distance z c  from the mirror is, under Fresnel or paraxial approximation,
 
 E   A ( x   c )=∫ dx   m   E   o   Q   z     o   ( x   m   −x   o ) Q   −f     A   ( x   m ) Q   z     c   ( x   c   −x   m )
 
 E   o   Q   z     o     −f     A   ( x   o ) Q   z     A     +zc ( x   c   −x   A )  [Equation 1]
 
where Q represents the quadratic phase function
 
                       Q   z     ⁡     (   x   )       ≡     exp   ⁡     [         ⅈ   ⁢           ⁢   k       2   ⁢           ⁢   z       ⁢     x   2       ]               [     Equation   ⁢           ⁢   2     ]               
and (x A , y A ; z A ) is the image position of the source point through the curved mirror:
 
                       x   A     =       -       f   A         z   o     -     f   A           ⁢     x   o         ;       z   A     =       -       f   A         z   o     -     f   A           ⁢     z   o                 [     Equation   ⁢           ⁢   3     ]               
For compactness of expressions, the overall propagation factors of the form exp(ikz) as well as the constant factors are ignored. In addition, and all (x, y) terms are abbreviated with (x) only. The wavelength is λ=2π/k. In these calculations, it is useful to note a basic property
 
 Q   z     1   ( x−x   1 ) Q   z     2   ( x−x   2 )= Q   z     1     +z     2   ( x   1   −x   2 ) Q   z     12   ( x−x   12 )  [Equation 4]
 
with
 
     
       
         
           
             
               
                 
                   
                     
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     The other mirror M B  with focal length f B  generates another output field E B (x c ) with similar definitions of x B  and z B . The two components superpose and interfere to produce the Fresnel zone-like intensity pattern recorded by the camera. One of the cross-terms is
 
 G   0 ( x   c   ;x   o )= E   A   E   B   *=I   o ( x   o ) Q   z     AB   ( x   c   −ax   o )  [Equation 6]
 
so that the intensity at the camera is
 
                     g   ⁡     (       x   c     ;     x   o       )       =                E   A     +     E   B            2     =     2   ⁢           ⁢         I   o     ⁡     (     x   o     )       [     1   +     cos   ⁢         k   ⁡     (       x   c     -     a   ⁢           ⁢     x   o         )       2       2   ⁢           ⁢     z   AB             ]                 [     Equation   ⁢           ⁢   7     ]               
where
 
                     α   =     -       z   c       z   o           ⁢     
     ⁢       z   AB     =     -         (       z   A     +     z   c       )     ⁢     (       z   B     +     z   c       )           z   A     -     z   B                     [     Equation   ⁢           ⁢   8     ]               
The point-source interference intensity g(x c ; x o ) has the familiar concentric ring pattern of radially-increasing frequency.
 
     For an extended object illuminated by incoherent light, the source points are incoherent with each other. Therefore, the total intensity is the sum or integral of the point-source interference intensities g(x c ; x o ) contributed by all the source points,
 
 h ( x   c )=∫ dx   o   g ( x   c   ;x   o )  [Equation 9]
 
which, as soon as a significant number of source points add up, quickly washes out the fringe structures. On the other hand, the phase-shifting process can be applied to extract the integral of the complex cross-term. As noted above, the mirror M A  is piezo-mounted so that a global phase φ can be applied. Then the interference intensity is
 
 h   φ ( x   c )=∫ dx   o   g   φ ( x   c   ;x   o )=∫ dx   o   |E   A   e   iφ   +E   B|   2   [Equation 10]
 
and a four-step phase-shift process yields
 
                             H   0     ⁡     (     x   c     )       =       ⁢       (     1   /   4     )     ⁢     {       [       h   0     -     h   π       ]     -     ⅈ   ⁡     [       h     π   /   2       -     h     3   ⁢           ⁢     π   /   2           ]         }                   =       ⁢     ∫       ⅆ     x   o       ⁢       G   0     ⁡     (       x   c     ;     x   o       )                       =       ⁢     ∫       ⅆ     x   o       ⁢       I   o     ⁡     (     x   o     )       ⁢       Q     z   AB       ⁡     (       x   c     -     α   ⁢           ⁢     x   o         )                       =       ⁢       I   o   ′     ⊙       Q     z   AB       ⁡     (     x   c     )                       [     Equation   ⁢           ⁢   11     ]               
where
 
                       I   o   ′     ⁡     (     x   c     )       -       I   o     ⁡     (       x   c     α     )               [     Equation   ⁢           ⁢   12     ]               
is the scaled object intensity pattern and the symbol ⊙ represents the convolution. The subscript 0 in G 0  and H 0  signifies the absence of aberration. That is, one obtains a complex hologram as a convolution of the object intensity I′ o  with a complex point spread function (PSF) Q z     AB   . For holographic reconstruction, the complex hologram H 0 (x c ) can be back propagated by the distance −z AB  
 
 I′   o ( x   c )= H   0   ⊙Q   −z     AB   ( x   c )  [Equation 13]
 
     Now consider IDH imaging in the presence of the phase aberrator Ψ(x′) at the Σ(x′, y′) plane. The field component A arriving at the output plane Σ c (x c , y c ) is now
 
 E   A ( x   c )=∫ dx′∫dx   m   E   o   Q   z* ( x′−x   o )Ψ( x ′) Q   z′ ( x   m   −x ′) Q   −fA ( x   m ) Q   z     c   ( x   c   −x   m )
 
= E   o   Q   z     o−fA   ( x   o ) Q   z     A     +z     c   ( x   c   −x   A )Φ A ( x   c   −α   A   x   o )  [Equation 14]
 
where
 
Φ A ( x )=[ψ⊙ Q   ζ     A   ](β A   x )=∫ dx ′Ψ( x ′) Q   ζ     A   ( x′−β   A   x )  [Equation 15]
 
and
 
                           α   A     =       ⁢         -       z   ′       z   ″         ⁢         z   A     +     z   ⁢           ⁢   c         z   A         +   α                   β   A     =       ⁢         z   ″         z   ″     +     z   ′         ⁢       z   A         z   A     +     z   c                         ζ   A     =       ⁢         z   ″         z   ″     +     z   ′         ⁡     [       z   ′     +         z   ″         z   ″     +     z   ′         ⁢         z   A     ⁢     z   c           z   A     +     z   c             ]                     [     Equation   ⁢           ⁢   16     ]               
Combining with a similar expression for E B , one can obtain the complex hologram of a single source point
 
 G   Ψ ( x   c   ;x   o )= E   A   E*   B  
 
= I   o ( x   o ) Q   z     o     −f     A   ( x   o ) Q*   z     o     −f     B   ( x   o ) Q   z     A     +z     c   ( x   c   −x   A ) Q*   z     B     +z     c   ( x   c   −x   B )Φ A ( x   c −α A   x   o )Φ* B ( x   c −α B   x   o )
 
= I   o ( x   o ) Q   z     AB   ( x   c   −αx   o )Φ A ( x   c −α A   x   o )Φ* B ( x   c −α B   x   o )  [Equation 17]
 
and for an extended object
 
 H   Ψ ( x   c )=∫ dx   o   G   Ψ ( x   c   ;x   o )
 
=∫ dx   o   I   o ( x   o ) Q   z     AB   ( x   c   −αx   o )Φ A ( x   c −α A   x   o )Φ* B ( x   c −α B   x   o )   [Equation 18]
 
     The last integral is turned into a convolution if z′=0, as described below, so that α A =α B =α. Then
 
 G   ψ ( x   c   ;x   o )= I   o ( x   o )[ Q   z     AB   Φ A Φ* B ]( x   c   −αx   o )  [Equation 19]
 
and
 
 H   ψ ( x   c )= I′   o   ⊙[Q   z     AB   Φ A Φ* B ]( x   c )  [Equation 20]
 
     Aberration compensation for adaptive optics proceeds as follows. A hologram of a guide star of unit magnitude at the center of the field is acquired to yield
 
 G   ψ ( x   c )≡ G   ψ ( x   c ;0)=[ Q   z     AB   Φ A Φ* B ]( x   c )  [Equation 21]
 
The full-field complex hologram H Ψ (x c ) is then
 
 H   ψ ( x   c )= I′   o   ⊙G   ψ ( x   c )  [Equation 22]
 
An attempt to reconstruct by numerical propagation Q −z     AB   =G* o  corresponds to
 
 I   ψ   ≡H   ψ   ⊙G*   0   =H   ψ     G   0   =I′   O   ⊙[G   ψ     G   0 ]  [Equation 23]
 
which in general does not yield accurate reproduction of the object I′ o . The symbol   stands for correlation. On the other hand, use of the guide star hologram leads to
 
 Î   ψ   ≡H   ψ     G   ψ   =I′   o   ⊙[G   ψ     G   ψ   ]≈I′   o   ⊙δ−I′   o   [Equation 24]
 
That is, as long as the auto-correlation of the guide star hologram is sharp enough, the result is an accurate reproduction of the object I′ o . In the absence of the aberration the delta function is exact, G 0   G 0 =δ. Equivalently, the hologram aberration can be compensated by subtracting the phase error of G Ψ  from H Ψ  in the Fourier domain. That is,
 
 Î   ψ   =H   ψ   ⊙G*   0   [Equation 25]
 
where
 
 Ĥ   ψ   =H   ψ   ( G   ψ     G   0 )  [Equation 26]
 
The latter method involves an additional correlation operation, but has the flexibility of being able to re-center the guide star hologram G 0  to avoid lateral shift of the final full-field reconstructed image. For comparison, one can also calculate the direct image formed when one of the mirror M B  focuses the image on the camera plane while the other mirror is blocked so that
 
 I   B ( x   c )=∫ dx   o   |E   B ( x   c   ;x   o )| 2   =∫dx   o   I   o ( x   o )|Φ( x   c   −αx   o )| 2   [Equation 27]
 
When focused, z B +z c =0 so that β B →∞ and ζ B →∞. However, closer examination of the exponent in Q ζ     B   (x′−β B x) shows that
 
|Φ′ B ( x   c )| 2 −|{tilde over (ψ)}( x   c )| 2   [Equation 28]
 
where {tilde over (ψ)} is the Fourier transform of Ψ
 
                       ψ   ~     ⁡     (     x   c     )       =     ∫       ⅆ     x   ′       ⁢     ψ   ⁡     (     x   ′     )       ⁢     exp   ⁡     [       -   ⅈ     ⁢           ⁢   k   ⁢       x   c       z   c       ⁢     x   ′       ]                   [     Equation   ⁢           ⁢   29     ]               
so that
 
 I   B ( x   c )=[ I′   o ⊙|{tilde over (ψ)}| 2 ]( xc )  [Equation 30]
 
As expected, the PSF is the absolute square of the Fourier transform of the aperture.
 
     A series of simulation studies were conducted to characterize the predictions of the above-described theory. IDHAO was experimentally implemented using the configuration depicted in  FIG. 2 , and the simulations assume a similar configuration. As shown in  FIG. 2 , a system  20  for performing aberration compensation using incoherent digital holography, i.e., an IDHAO system, generally comprises an optical system  22 , an interferometer  24 , and a computing system  26 . In the illustrated example, the optical system  22  is a telescope that is used to image an object field that includes a group of stars S. The optical system  22  is represented by an objective lens L o  and a relay lens L a  that together form an intermediate image in front of the interferometer  24 . As is shown in  FIG. 2 , a phase aberrator Ψ is placed close to the objective lens L o  on the side that faces the stars S. The aberrator Ψ represents, for example, the atmospheric turbulence in astronomical applications or inhomogeneities of a medium in microscopy applications. 
     The interferometer  24  includes a beam splitter BS and two mirrors M A  and M B . Mirror M A  is a planar mirror while M B  is a curved (concave) mirror. In some embodiments, the mirror M B  has a radius of curvature of approximately 1000 to 1200 mm and, therefore, a focal length f B  of approximately 500 to 600 mm. The mirror M A  is mounted to a linear actuator  28 , such as a piezoelectric actuator, that can adjust the position of the mirror along the optical axis for phase shifting. In some embodiments, the actuator  28  is capable of nanometer-scale adjustment of the mirror M A . The interferometer  24  further includes an imaging lens L c  that focuses the waves reflected by the mirrors M A  and M B  onto a camera  30 , which can comprise a CCD or other light detector. As is further shown in  FIG. 2 , the interferometer  24  can also comprise an interference filter IF positioned near the camera  30  that helps improve the interference contrast. 
     During use of the system  20 , the objective lens L o  of focal length f o  forms an intermediate image in front of the interferometer  24 . The relay lens L a  is used to image the input pupil onto the mirrors M A  and M B , achieving the requirement of z′=0. The imaging lens L c  is used, in combination with L o , to adjust the magnification and resolution of the system  20 . 
     With further reference to  FIG. 2 , the computing system  26  generally comprises a processing device  32  and memory  34  (i.e., a computer-readable medium) that stores an aberration compensation system  36  that includes one or more algorithms (i.e., logic). As is described below, image data, such as interference patterns, captured by the camera  30  can be provided to the computing system  26  for processing including the generation of aberration-compensated images of the object field. 
     The system  20  can be used to perform basic IDH. An example IDH process will now be described in relation to  FIG. 3 . In this example, the object field I o  is the Big Dipper, as shown in  FIG. 3( a ) , and it is assumed that there is no aberration for which to compensate. Each image frame in this example includes 256 pixels and is 10 mm across. The assumed optical configuration is such that the light from each point source of the object field arrives on the mirrors M A  and M B  as a plane wave. The mirror M B  is assumed to have a focal length f B  of 3000 mm, the imaging lens L c  is absent, and the camera is positioned at z c =z 4 +z 5 =1000 mm from the mirrors so that the holographic image distance is z i =2000 mm. The wavelength is set to λ=633 nm. 
     Light waves from the object field are received by the interferometer  28  from the optical system  22 . The light waves are divided by the beam splitter BS and copies are provided to each mirror M A  and M B . The mirrors M A  and M B  reflect the light waves so that they are received by the camera  30 . Because the mirrors M A  and M B  have slightly different curvatures, the waves that arrive at the camera  30  have different curvatures. The waves interfere with each other and, because of their differences in curvature, form a ring pattern. An example of such a ring pattern is shown in  FIG. 3( b ) , which is an example raw interferogram h φ . One ring pattern is formed for each point source of the object field, in this case each star of the Big Dipper (seven in total). 
     Multiple raw interferograms can be obtained with the mirror M A  in different positions to obtain multiple phase-shifted interferograms. For example, a first interferogram can be obtained when the mirror M A  is in a first position. The mirror M A  can then be moved approximately 1 to 650 nm along the optical path to a second position using the actuator  28  and a second interferogram can be obtained. This process can be repeated as many times as desired to obtain as many interferograms as desired. Although only two phase-shifted interferograms are necessary, many more can be obtained to improve results. In this example, it is assumed that four raw interferograms are obtained. 
     Once the raw interferograms have been obtained they can be combined by the computer system  26  to form a complex hologram H 0  of the object field.  FIGS. 3( c ) and 3( d )  respectively shown the amplitude and phase profiles of an example complex hologram. It is noted that, in most of the illustrations in this disclosure, complex images, such as complex holograms, are shown as a pair of amplitude and phase images. The amplitude images were rendered with the Matlab definition of “jet” color map, while for the phase images the blue-white-red color map corresponds to the range [−π,π]. Once the complex hologram has been obtained, numerical propagation to an appropriate distance, such as the distance to the intermediate image formed between the objective lens L o  and the relay lens L a , can be performed to obtain a reconstructed image.  FIGS. 3( e ) and 3( f )  show the amplitude and phase profiles of the reconstructed image, respectively. 
     In cases in which aberration is present, the system  20  can be used in an IDHAO context to compensate for the aberration. Such compensation can be, for example, performed by the aberration compensation system  36  of the computing system  26 . An example IDHAO process is described in relation to in  FIG. 4 . In this example, the Big Dipper is again used as the input image I o , as shown in  FIG. 4( a ) . The assumed phase aberration profile Ψ is shown in  FIG. 4( b )  and consists of two Zernike polynomial terms, ψ=a ψ (Z 3   +1 +Z 5   −1 ) with a Ψ =−0.5. As before, multiple phase-shifted interferograms can be obtained for the object field. In addition, multiple phase-shifted interferograms can also be obtained for a guide star, i.e., a single point source (a single star in this example) that can be used as a reference for compensation purposes. This can be achieved by obstructing much of the light from the object field using an aperture so that only light from the guide star is received by the interferometer  24 .  FIG. 4( c )  shows an example raw interferogram g φ  for the guide star and  FIG. 4( d )  shows an example raw interferogram h φ  for the full-field illumination (i.e., all of the stars). 
     As described above, multiple phase-shifted inteferograms can be obtained, both for the guide star and the full field. The inteferograms can then be combined to form a complex hologram G Ψ  for the guide star and a complex hologram H Ψ  for the full field.  FIGS. 4( e ) and 4( f )  show the amplitude and phase profiles of a guide star complex hologram that results from a set of four quadrature phase-shifted interferograms, and  FIGS. 4( g ) and 4( h )  show the amplitude and phase profiles of a full-field complex hologram that results from a set of four quadrature phase-shifted interferograms. 
       FIG. 4( i )  shows an expected direct image I B  through the aberrator without any holographic process, while  FIGS. 4( j ) and ( k )  show the reconstruction I Ψ  from the hologram H Ψ . As can be appreciated from these figures, both I B  and I Ψ  have serious degradation because of the aberration. Therefore, aberration compensation is needed.  FIGS. 4( l ) and 4( m )  show the amplitude and phase of the compensated hologram Î ψ . Note that the vortex-like structures in  FIG. 4( h )  have been replaced by more concentric ones in  FIG. 4( m ) .  FIGS. 4( n ) and ( o )  show the amplitude and phase profiles of an aberration-compensated image I ψ . The aberration-compensated image can be obtained by numerical propagation from the compensated hologram Î ψ  or by correlation of the guide star complex hologram G Ψ  ( FIGS. 4( e ) and 4( f ) ) and the full-field complex hologram H Ψ  ( FIGS. 4( g ) and 4( h ) ). In the correlation process, whenever a pattern within the complex hologram G Ψ  matches a pattern in the complex hologram H Ψ , a large peak results that coincides with the location of a point source of the object field. The stronger the correlation, the larger the peak and the greater the certainty of the location of the point source. Notably, this occurs irrespective of which point source is selected as the guide star as long as the mirrors M A  and M B  are the same distance from the relay mirror L a . That distance corresponds to the image plane of the objective lens L o . 
     As is apparent from  FIG. 4( n ) , the compensated image Î ψ  has much less distortion than the direct image I B  or the uncompensated holographic image I Ψ , and therefore provides a well-focused image of the point sources of I o . 
       FIGS. 5( a ) and 5( b )  describe an example method for compensating for aberration using incoherent digital holography that is consistent with the discussions of  FIGS. 3 and 4  above. Beginning with block  50  of  FIG. 5( a ) , light from an object field is received by an interferometer from an optical system, such as a telescope. While a telescope has been explicitly identified, it is noted that the optical system could take other forms. For example, the optical system could be that of a microscope or other imaging device. Turning to block  52 , the light waves that are reflected by the mirrors of the interferometer are captured. As described above, the waves can be reflected by a planar mirror M A  and a curved mirror M B  and can be captured by a camera, such as a CCD. In addition, the light waves received from the optical system can have been split by a beam splitter of the interferometer to provide copies of the waves to both mirrors. As described above, the waves reflected by the mirrors interfere with each other and form an interference pattern, which can be captured by the camera. A raw interferogram h φ  of the full object field can therefore be generated, as indicated in block  54 , and can be stored by the computing system. An example of a full-field interferogram is shown in  FIG. 4( d ) . 
     With reference next to decision block  56 , flow from this point depends upon whether a further interferogram is to be obtained. If only one interferogram has been obtained to this point, at least one more interferogram will be obtained. If more than one interferogram has already been obtained, however, flow depends upon whether how many interferograms have been obtained and how many are desired to for purposes of generating a full-field complex hologram. Assuming that another interferogram is to be obtained, flow continues to block  58  and the planar mirror M A  is displaced along the optical axis of the system for purposes of phase shifting. As described above, the distance that the mirror is displaced can be very small. By way of example, the mirror can be displaced approximately 1 to 650 nm. Such fine movement can be obtained using a precise actuator, such as a piezoelectric actuator. 
     Once the mirror M A  has been displaced, flow returns to block  50  and the above-described process is repeated so that a further full-field interferogram is generated. With reference again to decision block  56 , if the desired number of interferograms has been obtained, flow continues to block  60  at which the interferograms are combined to generate a full-field complex hologram H Ψ .  FIGS. 4( g ) and 4( h )  illustrate the amplitude and phase profiles of an example full-field complex hologram. 
     At this point, numerical propagation could be used to form a reconstructed image. If there is aberration present, however, that image would be distorted. Accordingly, aberration compensation should be performed. As described above, such compensation can be achieved by performing correlation between the full-field complex hologram and a complex hologram of a guide star selected from the full object field. In such a case, the full-field light can be obstructed so that only light from a selected guide star can be received, as indicated in block  62 . As described above, this can be achieved using an aperture that blocks light from other light sources within the object field. Once the full-field light is obstructed, light can be received from the guide star by the interferometer, as indicated in block  64 , and the light waves that are reflected by the mirrors of the interferometer are captured, as indicated in block  66 . As above, the wave reflected by the mirrors interfere with each other and form an interference pattern, which can be captured by the camera. A raw interferogram g φ  of the guide star can therefore be generated, as indicated in block  68  of  FIG. 5( b ) . An example of such a guide star interferogram is shown in  FIG. 4( c ) . 
     With reference next to decision block  70 , flow from this point depends upon whether a further interferogram is to be obtained. In some embodiments, the same number of interferograms that were obtained for the full field is obtained for the guide star. If another interferogram is to be obtained, flow continues to block  72  and the planar mirror M A  is displaced along the optical axis of the system for purposes of phase shifting. Once the mirror M A  has been displaced, flow returns to block  64  of  FIG. 5( a )  and the above-described process is repeated so that a further guide star interferogram is generated. Once the desired number of interferograms has been obtained, flow continues to block  74  of  FIG. 5( b )  at which the interferograms are combined to generate a guide star complex hologram G Ψ .  FIGS. 4( e ) and 4( f )  illustrate the amplitude and phase profiles of an example guide star complex hologram. 
     At this point, a complex hologram has been generated for both the full object field and the guide star. To compensate for aberration, mathematical correlation can be performed between the guide star complex hologram and the full-field complex hologram, as indicated in block  76 . In the example of  FIG. 4 , this means that  FIGS. 4( e ) and ( f )  are correlated with  FIGS. 4( g ) and ( h ) . As described above, in the correlation process, whenever a pattern within the complex hologram G Ψ  matches a pattern in the complex hologram H Ψ , a large peak results that coincides with the location of a point source of the object field. When such correlation is performed, an aberration-compensated image of the object field can be generated, as indicated in block  78 .  FIG. 4( n )  is an example of such an aberration-compensated image. 
     The behavior of IDHAO was simulated as several different parameters were varied, including the resolution, aberration strength, and aberration type. IDHAO was seen to be effective under wide range of these parameters, as well as under increasing noise added to the phase profile. Aberration compensation is expected to be equally effective for extended continuous objects as well as point-source objects. 
     As shown in  FIG. 6( a ) , the resolution of the IDHAO-reconstructed images is found to be consistent with the expected behavior of δx∝λf B /a, where a=10 mm is the frame size, the mirror M A  is planar, and the focal length of M B  is varied over a range. In  FIG. 6( b ) , an aberration ψ=a ψ (Z 3   +1 +Z 5   −1 ) with a Ψ =−0.5 is assumed. Evidently, the resolution of reconstructed image is not significantly affected by the aberration when compensated by IDHAO and behaves in an otherwise similar manner as in  FIG. 6( a ) . The image resolution is also expected to depend on the arrangement of the imaging lens L c  and the camera with respect to the overlap of the reflections from the two mirrors M A  and M B , though the imaging lens was not included in the simulation. 
     Behavior of the IDHAO with increasing strength of aberration is illustrated in  FIG. 7 . The aberration was varied for ( a ) a Ψ =−0.1, ( b ) a Ψ =−0.2, ( c ) a Ψ =−0.5, and ( d ) a Ψ =−1.0 of the same type ψ=a ψ (Z 3   +1 +Z 5   −1 ) as above. Other parameters were as in  FIG. 4 . The aberration phase profiles are displayed on the left of each row. Then, I Ψ  is the uncompensated holographic image and Î ψ  is the holographic image compensated by IDHAO, while the direct image I B  is also shown for comparison. That is, I Ψ  or I B  are the “before” images, while Î ψ  is the “after” image. It can be seen that the compensated image maintains reasonable quality for even high levels of aberration, whereas both the uncompensated and direct images deteriorate quite rapidly. For the strongest aberration in  FIG. 7( d ) , the aberration completely obliterates the uncompensated image I Ψ  and the direct image I B , though the latter develops a numerical artifact that makes the obliteration less obvious. However, the compensated image Î ψ  still maintains good resolution and contrast. On the other hand, note that while the compensated image Î ψ  maintains relatively sharp central peaks, the peak height does decrease and a broader halo develops around each peak as the aberration increases. 
     The IDHAO process is equally effective for various types of aberrations, as shown in  FIG. 8 . In particular, the effectiveness of IDHAO is not contingent upon the symmetry of the aberration profile. Note that the uncompensated images I Ψ  and I B  show similar patterns of distortion for each type of aberration, and the compensation removes most of the distortions in Î ψ . 
     Next, in  FIG. 9 , a gray-scale photographic image is used as the model object, most other parameters being the same as before. Because of the large number of non-zero object pixels, the computation is substantially longer compared to the preceding simulations, where the object consisted of seven illuminated pixels. The IDHAO process needs to be calculated for each source point before integrating over all pixels. However, the IDHAO process is seen to behave as expected and is able to remove much of the distortion and mostly recover the resolution loss due to the assumed phase aberration. 
     Lastly, the behavior of IDHAO in the presence of noise was considered. In  FIG. 10 , the noise was simulated by adding random phase noise on the complex holograms G Ψ  and H Ψ   0  to each pixel in the holograms is added a phase of a φ  times a random number in the range [0,1], with a φ =5 for rows ( a ) and ( c ) or a φ =10 for rows ( b ) and ( d ). The first two rows ( a ) and ( b ) assume no aberration, a φ =0, while the lower two rows ( c ) and ( d ) assume a φ =0.2, in ψ=a ψ (Z 3   +1 +Z 5   −1 ). The phase noise in each case is significant enough to almost obscure the underlying Fresnel ring patterns, though not completely. Reconstructed images I Ψ  without compensation suffer and, especially when a small amount of aberration is present, the noise almost completely obliterates the signal. On the other hand, the compensated images Î ψ  degrade only mildly even when both phase noise and aberration are present. This behavior is remarkable but also reasonable in view of the fact that Î ψ  is the correlation of H Ψ  with G Ψ , meaning that any noise or aberration in H Ψ  that has low correlation with G Ψ  does not have significant contribution to the final image Î ψ . 
     Note that this simulates noise in the detection system, not in the optical field itself. Noise in the optical field is accounted for as aberration if it is present at a plane conjugate to the interferometer mirrors. Noise or aberration away from the conjugate plane is not compensated by the present scheme of IDHAO, unless one sets up a multi-conjugate configuration, which may be possible borrowing some of the techniques available in conventional AO systems. 
     A series of experiments were performed to confirm the results of theoretical and simulation studies and to observe general behavior of the IDHAO process. First, to demonstrate the basic IDH with focusing property, the optical configuration of  FIG. 2  was set up using a group of three red light emitting diodes (LEDs) at z 1 =650 mm and another red LED closer at z   1 =440 mm. All three lenses were of focal length f o =f a =f c =100 mm, while the curved mirror had a focal length of f B =600 mm. Various distances were set at z 2 =220 mm, z 3 =200 mm, z 4 =200 mm, and z 5 =140 mm, ±5 mm. A Pixelfly CCD camera having a 6.3×4.8 mm 2  sensor area with 640×480 pixels was used. This area projects to approximately a 60×45 mm 2  field of view at the object plane at z 1  from L o . The plane mirror M A  was piezo-mounted and driven with a sawtooth output of a digital function generator and four phase-shift interferograms were acquired at 20 fps. Hologram acquisition, reconstruction, and aberration compensation were performed using LabVIEW-based programs. The entire cycle took a few seconds, including file write/read and without any attempt to optimize the speed. 
     One of the raw holograms h φ  of the four-LED object field is shown in  FIG. 11( a )  and the corresponding complex hologram H 0  is shown in  FIG. 11( b ) . One may note the slightly higher fringe frequency of the lower left LED compared to the other three LEDs because of its closer distance to the objective lens. Numerical propagation of the hologram focuses the LEDs at two separate distances, ( c ) at z i =22000 mm and ( d ) at z i =14870 mm, which values are consistent with the effective image distances projected to the object space. The image distances can be calculated by using Eq. (8) and geometrical optics consideration, and the approximately 10:1 lateral magnification corresponds to 100:1 longitudinal magnification. 
     Phase aberrators were fabricated by coating optically clear plastic piece with acrylic paint to produce irregular surface profiles of varying degrees. An aberrator was placed just behind the objective lens L o , and the resulting IDH images are shown in the upper row of  FIG. 12 . The concentric rings of  FIGS. 11( a ) and ( b )  were replaced with severely distorted patterns in  FIGS. 12( a ) and 12( b ) . An attempt to numerically focus the images from H Ψ  yielded the results shown in  FIGS. 12( c ) and 12( d )  at the two expected image distances. A star hologram G Ψ  was acquired by aperturing the object field to leave only one of the LEDs on, in this case one belonging to the three-LED group at z 1 =650 mm, as shown in  FIG. 12( e ) . Aberration compensation using the guide star hologram, as described in theory and simulation, resulted in the corrected hologram Ĥ ψ  in  FIG. 12( f )  and the corrected images Î ψ  in  FIGS. 12( g ) and ( h ) . Note, as in the simulation of  FIG. 4 , that the distorted fringes in  FIG. 12( b )  are replaced with more concentric ones in  FIG. 12( f ) . Note also that the guide star at z 1  is effective in compensating the images at both z 1  and z   1 . Another test, not presented, of using a guide star at z   1  was equally effective in compensating the images at the two distances. That is, a guide star can be placed at any distance relative to the object volume. This is a unique property of IDHAO, which may not be available in other existing adaptive optics techniques. 
     Next, the LED brightness was reduced to various levels to observe effects of noise and low signals on the IDH and IDHAO images. In  FIG. 13 , all four LEDs were placed at z 1 =650 mm, which constitutes the object field. The leftmost LED, was also used as the guide star. No deliberate aberrator was in place. In each row of  FIG. 13 , the first image, column ( i ), is the reconstructed image I Ψ  without compensation, while columns ( ii )-( iv ) are the compensated images Î ψ  with increasing brightness of the guide star. The rows ( a )-( c ) represent the object field with decreasing brightness. In all cases, the IDHAO process is seen to be capable of reducing the noise and increasing the contrast. Also significant to note is that the contrast enhancement increases with the guide star brightness. The effect is especially marked for the weakest signal in ( c ), where the uncompensated image is almost buried in background, but the brighter guide star brings out the signal with high contrast, while maintaining good resolution. 
     In addition to the point-like sources of LEDs, extended objects of finite surfaces were also used to test the IDHAO process. As shown in the top row of  FIG. 14 , a miniature halogen lamp illuminated a resolution target from behind through a piece of frosted glass, while in the bottom row, a knight chess piece was illuminated with the halogen lamp in front. The objects were at z   1 =450 mm. In both cases an interference filter, centered at 600 nm and of linewidth 10 nm, was placed in front of the camera to improve the interference contrast, although no significant difference in resulting images was noticed. In each case, a raw interferogram h φ  and the complex hologram H Ψ  are shown. Note that for these extended objects the raw interferograms only have barely recognizable interference contrast. No deliberate phase aberrator was present. An LED placed next to the objects at z   1 =450 mm served as the guide star. 
     With reference next to  FIG. 15 , an experiment similar to that of  FIG. 13  was carried out by varying the brightness of the halogen lamp and the guide star LED to various levels. Again, it was seen that the IDHAO process was quite effective in extracting and strengthening the image signals buried in background noise. 
     Finally,  FIG. 16  shows that the IDHAO process is effective for a three-dimensional object volume, as was found for LED objects in the experiment of  FIG. 12 . In this experiment, an LED at z 1 =650 mm was turned on in addition to the knight piece at z   1 =450 mm. The guide star LED was still at the same distance as the knight. In the corrected images of the bottom row, as well as the uncorrected images of the top row, the objects separately came into clear focus at the appropriate distances. 
     The disclosed incoherent digital holographic adaptive optics (IDHAO) method is an effective and robust means of compensating optical aberrations in incoherent optical imaging systems. A theoretical framework based on propagation of spherical waves from source points followed by integration of intensity over the object points correctly predicts the possibility of aberration compensation. General behavior of the theoretical model of IDHAO was investigated by simulation studies. The studies showed robust aberration compensation under various ranges of parameters, including aberration types and strength, as well as in presence of significant noise. Finally, experimental studies corroborated these predictions. The IDHAO process was seen to be effective for both point-like sources and extended objects. In particular, a brighter guide star is seen to be very effective in bringing out weak signals buried under noise, through the fact that the IDHAO signal is proportional to the auto-correlation of the complex guide star hologram.