Phase retrieval from a high-numerical-aperture intensity distribution by use of an aperture-array filter

Abstract

Almost all noninterferometric phase-retrieval methods used in coherent diffractive imaging have been based on the measurement system with low numerical aperture, in which Fresnel or Fraunhofer approximation is valid to express the wave propagation between an object and a detector. In microscopy, which is a typical application of coherent diffractive imaging, the measurement of the diffraction intensity with high numerical aperture is required for object reconstruction at high spatial resolution. We here propose an extension procedure to apply the previous phase-retrieval method using an aperture-array filter [J. Opt. Soc. Am. A25, 742 (2008)] to the system with high numerical aperture, in which the first Rayleigh-Sommerfeld integral for spherical waves is utilized instead of the Fresnel integral for parabolic waves. Computer-simulated examples in the high-numerical-aperture system demonstrate object reconstruction at high lateral resolution and retrieval of information in the depth direction of an object.

Coherent diffractive imaging beyond the Fresnel approximation using a deterministic phase-retrieval method with an aperture-array filter

“…These results demonstrate the effectiveness of the extension procedure in Subsection 2.B. The maximum lateral resolution of the present system is given by δ l λ∕2 NA≅ 2.58 μm [15], which is about 10 times the resolution of the previous experiment [14] with the same array filter as in the present system. Thus the two-point resolution criterion in the object plane is obtained by twice the value (5.16 μm) of δ l .…”

“…On the other hand, there may be a problem in directly applying the iterative phase-retrieval algorithms, such as Fienup's algorithm [17], to the object reconstruction from a single 2D diffraction distribution intensity beyond the Fresnel approximation, because the algorithms are based on the use of one Fourier transform calculation between the object and its diffracted wave function. The one-to-one relationship between the object plane and its diffraction plane beyond the Fresnel approximation can be represented by using two consecutive 2D Fourier transforms with the diffraction kernel for spherical waves [15]. However, the convergence property of the iterative algorithms with the double Fourier transforms in the phase retrieval from only a single diffraction intensity distribution beyond the Fresnel approximation is not well defined in the literature yet, to the best of my knowledge.…”

“…As described in the previous paper [15], the NA at the limit of the Fresnel approximation can be defined by…”

“…For such a high resolution, we have to use not the Fresnel diffraction integral for parabolic waves but a rigorous diffraction integral for spherical waves [e.g., the first Rayleigh-Sommerfeld (RS) integral]. Thus we have previously proposed [15] an extension procedure to apply our method to the coherent diffractive imaging based on the exact diffraction formula, and then presented the effectiveness of the procedure by means of computer simulations. In this paper, we experimentally demonstrate at optical wavelengths that our phase-retrieval method with the extension procedure is valid for the object reconstruction from a single diffraction intensity measured at a high NA beyond the Fresnel approximation.…”

Phase retrieval based on the vectorial model of point spread function

“…The first and natural solution approach to high-NA phase retrieval is to adapt the scalar diffraction theory for high-NA imaging systems [25]. The main modification is that the additional defocus term used to calculate the corresponding out-offocus image in the low-NA settings is replaced by an appropriate spherical cap in the high-NA settings [25][26][27][28], see Sections 2A&2B. For higher-NA imaging systems (NA ≥ 0.6), however, the accuracy of this phase retrieval approach is limited due to model deviations.…”