Aberrations  Aberrations of Lenses  Analogue to Holographic Model  Aberrations in Holography  Implications in the aberration equations  Experimental Demonstration

Lenses image on-axis points perfectly. Non-paraxial and extended images exhibit aberrations.

An aberration in an imaging system is defined as any distortion of an image due to imperfections in the imaging system. In standard optics, these occur for non-axial object points and extremal rays. Note that a mis-aligned optic effectively creates an off-axis point.

There are 5 standard aberrations known as the Seidel Aberrations: o Spherical Aberration o Coma o Astigmatism o Petzval (Field)(Curvature o Distortion

Spherical Aberration

Coma

When sagittal rays focus, transverse rays make a horizontal line. When transverse is in focus, sagittal rays make a vertical line.

Field curvature

Distortion is due to magnification being different for the axial and marginal rays (This shows the stretching of a flat image – not a 3D bulging view)

The wave model compares the aberrated focus to an ideal spherical focus which is defined by the physical optic.

W = 0 C 40 (r 4 ) + 1 C 31 (  )(r 3 )cos(  ) + 2 C 22 (  ) 2 (r 2 )cos 2 (  ) + 2 C 20 (  ) 2 (r 2 ) + 3 C 11 (  ) 3 (r)cos(  ) Each of the the five terms in the aberration function represent the five Seidel aberrations: The first three terms lead to a blurring of the image point and the next two lead to a distortion of the image.

The Holographic Model In order to show a correlation between holography and conventional lenses, an equivalent “focal length”, f R for the hologram may be derived from the reconstructed phase wavefront. This is allowable insofar as the reconstructed phase wavefront is also a converging spherical wavefront and maybe thought of as the “reference sphere”.

Analogue of Seidel aberrations for Holography: Assume a point-source object, and compare the aberrated wavefront to the correct wavefront.

1.Assume a point source for the object and a point source for the reference. 2.Assume a different point source for the reconstruction source. 3.An image point is derived from the coordinates of the original object point, the original reference point and the alternate reference point. 4.Seidel aberrations are determined by the phase mismatch between the image point from the correct reference and the image point from the alternate reference point.

The Derivation: The emerging reconstructed phase wavefront is the sum of the recorded phases and the phase of the reconstruction wave.  =  c +/-  r +/-  o Signs indicate original (virtual) image or conjugate (real). The general expression from a point G for a spherical wave is  = 2  {[(x - x g ) 2 + (y-y g ) 2 + z g 2 ] 1/2 – [x g 2 + y g 2 + z g 2 ] 1/2 } Substituting this into the above, with the specific coordinates and replacing the general coordinates, an exact expression is obtained for the phase at the hologram plane.

This analysis now creates an image point derived from the coords of the original ref, the object and the recon beam and is given by image coordinates X i, Y i and Z i X i =(x c z o z r +  x o z c z r -  x r z c z o )/(z o z r +  z c z r -  z c z o ) Z i = z o z c z r /(z o z r +  z c z r -  z c z o )

 Thus, for the object beam coordinate, zo, and the equivalent image position Zi, under the assumption of a spherical wavefront, we may construct a lens equivalent expression  (1/z o ) + (1/Z i ) = 1/ f R ) = 1/z o +/- (  )(1/z o + 1/z r  by comparing the standard lens equation 1/s i + 1/s o = 1/f  and also Champagne's derivation of image distance 1/R f = 1/R c +/- (  )(1/R o + 1/R r )

Resolution Not valid to compare with electronic pixels. Strictly, the resolution of the image as a function of source size needs to be calculated using the Van Cittert-Zernike theorem May be described as a comparison of the size of the reconstruction point to the size of the image source, giving: dX i /d x c = (z o z r )/(z o z r +  ( z o z r – z o z c )) dZ i /d z c = (z o z r )/(z o z r +  (z c z r – z c z o ))

Magnification o Transverse o Longitudinal o Angular M T = dX i /dx o = 1/(1+/- z o /  z c – z o /z r ) (paraxial) M T = (cos  o /cos  i ){1/(1+/-z o /  z c – z o /z r )} (nonparaxial) M L = dZ i /dz o = -(1/  ){1/(1-z o [(1/  z c ) + (1/z r )} 2 -(1/  )MT M A = d(X i /Z i )/d(x o /z o ) =  (paraxial) M A =  cos  o /cos  i ) (nonparaxial)

Expanding the spherical wavefront expression to third order produces the third order aberrations – the Seidel Aberrations for holography. Once again, the difference between the ideal sphere and the actual wavefront, W, gives the actual aberratiion function.

W = 2  c [(-1/8)   S Spherical Aberration +(1/2)    C x cos  C y sin  Coma -(1/2)    x cos 2  A y sin 2  A xy cos  sin  Astigmatism +1(1/4)   F Petzval Curvature +(1/2) r(D x cos  + D y sin  )] Distortion Transforming to a circular coordinate system, this gives for the Seidel aberration function:

Experimental results A planar image was recorded at 514nm with a collimated reference at 30 degrees.

This was reconstructed under a variety of alternate beam geometries: o Varying reconstruction angle + /- o Varying beam divergence + / - o Alternate wavelength at those geometries We shall see how these changes affect the image by imaging onto a screen with a calibrated scale.

Lateral Displacement and Coma appear when the reconstruction angle is off.

Magnification occurs both laterally and longitudinally when the divergence is off.

When the angle and divergence are off, the image rolls to the side, exhibiting multiple aberrations.

When reconstructed at 633nm, all of these effects are further shifted… Image is displaced, shrunken, bowed, and cannot be focused

Conclusion: Best practice is to always reconstruct a hologram with a source that matches the construction geometry, including avoiding aberrations occurring within the reconstruction source itself. When aberrations occur, understanding the different effects will aid in troubleshooting the system.