JEOS RP ISSN03 | Seite 139

132

J. Eur. Opt. Society-Rapid Publ. 22, 14( 2026)

in a numerically exact solution of the Rayleigh – Sommerfeld diffraction integral [ 25 ] for situations where the destination pixel pitch is larger than the source pixel pitch. In this work, we propose a computational reconstruction method that also implements a numerically exact solution of the Rayleigh – Sommerfeld diffraction integral, but for the inverse case where – in the near field – the destination pixel pitch is smaller than the source pixel pitch.

2 Methods

2.1 Sampling and Fresnel propagation formula

Let us consider a typical lensless microscopy sampling scenario, such as the one depicted in Figure 1. Wefind the object domain on the left and parallel to it, at a distance z, the sensor domain. The sensor has a pixel pitch of Dp, which is usually several times larger than the optical resolution d of the system, so that Dp = q d with q > 1. If we denote the number of pixels along one edge by N, we can determine theedgelengthofthesensordomainbyL S = N Dp and the edge length of the object plane by L O = N d. Hence, the sensor domain is much larger than the object domain, whereas the resolution of the object domain is much higher than that of the sensor domain. We can now set all parameters in relation to each other if we demand the system to be diffraction limited. In this case, the optical resolution d is defined by the Abbe diffraction limit

d ¼ k 2 NA k z; ð1Þ

L S

with the numerical aperture NA = n sin( a) andthewavelength k. In the last step of equation( 1) we have assumed the paraxial approximation sin( a) tan( a)= L S /( 2z).

If light reflected or scattered by a specimen in the object plane is superposed with a reference wave R( x, y), the sensor will record an interference pattern

I ðx; yÞ ¼jU H j 2 þjRj 2 þ U H R þ U H R; ð2Þ

where U H( x, y) denotes the complex hologram, i. e., the object wave across the sensor domain. We partly omitted the function arguments for the sake of readability. We can use any phase shifting method to extract the term M( x, y)= U H( x, y) R *( x, y) from which we can obtain the complex hologram U H( x, y) ifthestructureofR is known.

To reconstruct the object wave field U O from the complex hologram U H we need to solve the inverse Rayleigh – Sommerfeld diffraction integral. Within the scalar diffraction theory, it provides an exact description of light propagation from a hologram plane x, y into an object plane u, v [ 26 ]:

U O ðu; vÞ ¼ 1 Z exp U H ðx; yÞ ð ikr Þ z

2p r r ik � 1

dxdy; r ð3Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where r ¼ ðx � uÞ 2 þðy � vÞ 2 þ z 2 is the distance between two points in the hologram plane and the object plane.

Fig. 1. Typical sampling scheme in lensless microscopy: The sensor plane and the object plane are separated by a distance z. The pixel pitch Dp in the sensor plane is larger than the pitch( resolution) d in the object plane. In the illustrated example, the factor between the grids is q = 4. The space bandwidth product of the system, i. e., the number of pixels N, is invariant. Consequently, the edge length of the sensor domain L S is larger than that of the object domain L O by a factor of q.

Due to the finite aperture of the complex hologram U H the resolution of even the analytically exact solution for the reconstructed object wave field U O that equation( 3) provides will always be limited by diffraction. We describe such a reconstruction as diffraction-limited. Consequently, a numerically exact solution to the inverse Rayleigh – Sommerfeld diffraction integral corresponds to a diffraction-limited reconstruction of the object wave field U O( u, v).

Initially, the Fresnel propagation formula readily comes to mind to solve equation( 3). It is based on the Fresnel approximations that assume r z outside the exponentials, ik�1 / r ik, andafirst order binomial approximation of r within the exponentials

r z þ 1 2z ðx � uÞ2 þðy � vÞ 2: ð4Þ

Its inverse propagates e. g., a measured wave field in the hologram plane U H( x, y) back into the object plane, as derived in [ 25 ], such that

U O ðu; vÞ ¼ikz S

2 ðu; vÞF �1 U H ðx; yÞS

1 ðx; yÞ u kz; v

; kz ð5Þ