Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region

Abstract

Abstract:The forward problem of focusing light using a high numerical aperture lens can be described using the Debye-Wolf integral, however a solution to the inverse problem does not currently exist. In this work an inversion formula based on an eigenfunction representation is derived and presented which allows a field distribution in a plane in the focal region to be specified and the appropriate pupil plane distribution to be calculated. Various additional considerations constrain the inversion to ensure physicality and practicality of the results and these are also discussed. A number of inversion examples are given.

Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system

“…Our expansion therefore provides a simple and natural way to carry out both forward and inverse analysis of high NA focusing systems. In contrast to our earlier work [9] which allowed only for one-dimensional (1-D) apodization techniques [12], our current expansion could be used to implement twodimensional (2-D) apodization and masking techniques to synthesize fields at the focal region of a high NA focusing system [13].…”

Fields in the Focal Space of Symmetrical Hyperboloidal Focusing Lens

“…The electric field distribution around focal point F of hyperboloidal lens, which has been excited by an electromagnetic plane wave polarized in x-direction and propagating in z-direction as shown in Figure 2, may be obtained using the Debye-Wolf focusing integral [36][37][38][39][40][41][42] and is given below (20) where k = position vector OP and T is the solid angle associated with all the ray which reaches the image space through exit pupil of the lens. The coordinates of the point P (ξ, η, ζ) on lens are defined in (4) and the co-ordinates (x, y, z) of a point F in the image region may be expressed in the form…”

“…Hongo and Ji [11][12][13][14][15][16], Naqvi and coworkers [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. Many investigations on the fields in focal space of focusing system have been carried out using different methods [33][34][35][36][37][38][39][40][41][42].…”

“…Sherif and Török [38][39][40][41][42] reported an eigenfunction representation of the integrals of Richards and Wolf. It is found that Debye-Wolf focusing integral and Maslov's method are of comparable accuracy.…”

Third-harmonic generation microscopy with Bessel beams: a numerical study

“…Indeed, the detected signal results from the coherent superposition in the detection plane of waves originating from different locations near focus, and the visibility of a particular distribution of scatterers is determined by interference phenomena. Engineering the spatial distribution of the intensity, phase and polarization of the focused driving field is possible by controlling the field at the pupil of the objective [8][9][10]. In coherent nonlinear imaging, focus shaping generally results in a modulation of phase matching and far-field scattering patterns [11][12][13], which potentially provides information about the sample microstructure.…”