Speckle imaging and hidden phase

The Knox-Thompson, or cross-spectrum, method provides two two-dimensional d ifference equations for the phase of the object spectrum. We demonstrate th at, in general, the object spectrum phase can be decomposed into a regular, single-valued function determined by the divergence of the phase gradient, as well as a multivalued function determined by the circulation of the pha se gradient; this second function has been called the hidden phase. The sta ndard least-squares solution to the two-dimensional difference equations wi ll always miss this hidden phase. We present a solution method that gives b oth the regular and the hidden parts of the object spectrum phase. Finally, we illustrate several examples of imaging through turbulence and postproce ssing with the Knox-Thompson method, including the hidden phase. (C) 2000 O ptical Society of America.