PSF Generator

This model describes the vectorial-based diffraction that occurs in the microscope [9, 10]. The phase aberration W(Θ) is given by the Gibson and Lanni model, but the three electric field components are evaluated independently. The PSF of this model is shift invariant in the lateral directions only. This model is defined by the following equation:

Spatial section XY	Axial section YZ
Running time: 12.53 ms to generate a 256x256x64 stack.

This model describes the scalar-based diffraction that occurs in the microscope [7]. It accounts for the immersion, the cover-slip and the sample layers. The PSF of this model is shift invariant in the lateral directions only. This model is defined by the following equation:

Spatial section XY	Axial section YZ
Running time: 3.92 s to generate a 256x256x64 stack.

This model is an extension to the Gibson and Lanni model, that describes the scalar-based diffraction that occurs in the microscope [10]. It captures the aberration that caused due to variation of refractive index within the thick specimen. This model captures shift varying nature of PSF in axial direction only by three predefined functions: linear Logarithmic, and Exponential. This model defined by the following equations:

where:

• x = (x, y, z) denotes the coordinate of a point in the image plane.
• x_p = (x_p; y_p; z_p) is the coordinate of the point source.
• τ = (NA; n; t) is the parameter set of the imaging system in which n is the refractive index and t is the thickness.
• r² = (x - x_p)² + (y - y_p)² is the point at which ray intersects the exit pupil.
• J₀ is the Bessel function of first kind of order zero.



Parameter	Description
NA	Numerical Aperture of the objective lens. This value is given by NA = ni sin(Θ) where Θ is one half of the angular aperture of the lens.
ns	Refractive index of the sample layer.
ts	Axial location of the point source within the sample layer. The corresponding input field in the plugin is "z".
ng	Actual value of the refractive index of the coverslip. This value is assumed to be equal to ng* and it is therefore not used by the plugin.
ng*	Nominal value of the refractive index of the coverslip. This value is assumed to be equal to ng and it is therefore not used by the plugin.
ni	Actual value of the refractive index of the immersion layer (the paper by Gibson and Lanni uses the notations noil).
ni*	Nominal value of the refractive index of the immersion layer (the paper by Gibson and Lanni paper uses the notations noil*. This value is assumed to be equal to ni and it is therefore not used by the plugin.
tg	Actual value of the coverslip thickness. This value is assumed to be equal to tg* and it is therefore not used by the plugin.
tg*	Nominal value of the coverslip thickness. This value is assumed to be equal to tg and it is therefore not used by the plugin.
ti	Actual working distance between the objective and the coverslip (the Gibson and Lanni paper uses the notation toil).
ti*	Nominal working distance between the objective lens and the coverslip (the Gibson and Lanni paper uses the notation toil*). This value is given by ti* = ti + Δ where Δ is the stage displacement. When generating a z-stack, stage displacement values are determined by the "Axial resolution" and by the "slices" input parameters of the plugin. If the number of slices is odd, then the middle slice corresponds to Δ = 0.
λ	Wavelength of the light emitted by the point source.
k	Wavenumber in vacuum of the emitted light, k = 2Π/λ.
xd, yd	Lateral position for evaluating the PSF at the detector plane. The sampling points are {xd,n, yd,m} = {nΔ, mΔ}n,m where Δ is the value of the "Pixel size in object space" input parameter, n = - (N-1)/2 ... (N-1)/2 and m = - (M-1)/2 ... (M-1)/2. N and M are the "width" and the "height" input parameters, respectively of the plugin. If N and M are odd, then the middle pixel is (0,0).
zd	Axial distance between the detector and the tube lens. This value is assumed to be equal to the focal length of the tube lens, and for this reason the plugin substitutes NA = a/zd in h(x, y, z).
W	The Gibson and Lanni phase aberration.
C	A normalizing constant. The value C/zd that appear in h(x, y, z) is determined by the "normalization" input parameter of the plugin.

Spatial section XY	Axial section YZ
Running time: 3.92 s to generate a 256x256x64 stack.

This model describes the scalar-based diffraction that occurs in the microscope when the particle is in focus [5, 6]. The imaging plane, however, need not be in focus. The PSF of this model is shift invariant in all directions. This model is defined by the following equation:

Spatial section XY	Axial section YZ
Running time: 942 ms to generate a 256x256x64 stack.

This models simulates defocusing in the Fourier domain by a modulated Gaussian function. The variance of Gaussian (σ) changes linearly with the axial axis. ω is the radial frequency which is defined by: ω² = ω_x² + ω_y². This model is defined by the following equation:

Spatial section XY	Axial section YZ
Running time: 2300 ms to generate a 256x256x64 stack.

This model simulates the blurring effect with a 2D gaussian function. The variance of Gaussian (σ) changes linearly with the axial axis. This model is defined by the following equation:

Spatial section XY	Axial section YZ
Running time: 1800 ms to generate a 256x256x64 stack.