Manuel Tessmer and Herbert Gross, "Generalized propagation of light through optical systems. II. Numerical implications," J. Opt. Soc. Am. A 32, 2276-2285 (2015)

We present an algorithm implemented in a MATLAB toolbox that is able to compute the wave propagation of coherent visible light through macroscopic lenses. The mathematical operations that complete the status at the end of the first paper of this sequence, where only limited configurations of the propagation direction were allowed toward arbitrarily directed input beam computations, are provided. With their help, high numerical aperture (NA) field tracing is made possible that is based on fast Fourier routines and is Maxwell exact in the limit of macroscopic structures and large curvature radii, including reflection and transmission. Whereas the curvature-dependent terms in the Helmholtz equation are under analytical control through the first perturbation order in the curvature, they are only included in the propagation distance in the current investigation for the sake of reasonable time consumption. We give a number of examples that demonstrate the strengths of our approach, describe essential differences from other approaches that were not obvious when Paper 1 was written, and list a number of drawbacks and possible simplifications to overcome them.

David Kessler and Roland V. Shack Appl. Opt. 31(15) 2692-2707 (1992)

References

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Intensities for a “Perfect” Asphere with Focal Length of $f=4.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ (Sphere Has Equivalent Radius of Curvature of $R=1.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$)^{a}

W

${I}_{y}/{I}_{x}$

${I}_{z}/{I}_{x}$

0.75

$0.0007\mathrm{e}-7$

$9.0979\mathrm{e}-4$

$1.4846\mathrm{e}-7$

$9.0126\mathrm{e}-4$

0.50

$0.0010\mathrm{e}-7$

$4.0178\mathrm{e}-4$

$0.2920\mathrm{e}-7$

$3.9966\mathrm{e}-4$

0.25

$0.1309\mathrm{e}-7$

$0.7339\mathrm{e}-4$

$0.0183\mathrm{e}-7$

$0.9991\mathrm{e}-4$

First lines correspond to our algorithm and the second ones to the RWI formulation. We omitted small input fields because, as we shrink the input size, the stitched beams strongly diverge, resulting in an unreasonable computational cost. We advise other approximations for the limit of small input distributions. Note that for high NA the effects of reflection grow in importance—a fact that may explain the behavior of the $y$ component (and also the $z$ component for more extreme situations that we did not consider here).

Table 2.

Output Data for a Uniform Circular Beam of Diameter ${d}_{\mathrm{beam}}=0.375\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$, Embedded on a Quadratic Area with Length of ${d}_{x}=0.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$^{a}

Polarized solely in $x$ direction (the very small $z$ component properly calculated as well), having a wavelength of 500 nm passing an aspherical surface to a medium with ${n}_{2}=1.8$. The focal length of the asphere is therefore $f=3.375\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$. There are variations in the results that also depend on the spatial magnification factor of the image and resulting interpolation effects implicitly generated by the inverse Fourier transformation.

Values for the paraxial lens are chosen in such a way that a circular Gaussian distribution at the object will be transformed into an elliptical one and to make the rays collimated again after the paraxial element, where only the $y$ direction is modified (for a case where the user likes to propagate an elliptical profile). The quantities $d$, $A$, and $B$ should therefore obey the law of paraxial optics to stretch the beam in the $y$ direction by a demanded value. For the sake of simplicity we also did the following. To get the normalized coordinate=coordinate itself: Set the entrance pupil diameter equal to 2. To get Gaussian distribution with${\mathrm{\Delta}}_{x}=0.25$: choose Gaussian apodization in the general menu and an apodization factor of $\mathrm{G}=16$. To cut off edge rays: surface 3 properties → elliptical aperture.

Tables (3)

Table 1.

Intensities for a “Perfect” Asphere with Focal Length of $f=4.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ (Sphere Has Equivalent Radius of Curvature of $R=1.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$)^{a}

W

${I}_{y}/{I}_{x}$

${I}_{z}/{I}_{x}$

0.75

$0.0007\mathrm{e}-7$

$9.0979\mathrm{e}-4$

$1.4846\mathrm{e}-7$

$9.0126\mathrm{e}-4$

0.50

$0.0010\mathrm{e}-7$

$4.0178\mathrm{e}-4$

$0.2920\mathrm{e}-7$

$3.9966\mathrm{e}-4$

0.25

$0.1309\mathrm{e}-7$

$0.7339\mathrm{e}-4$

$0.0183\mathrm{e}-7$

$0.9991\mathrm{e}-4$

First lines correspond to our algorithm and the second ones to the RWI formulation. We omitted small input fields because, as we shrink the input size, the stitched beams strongly diverge, resulting in an unreasonable computational cost. We advise other approximations for the limit of small input distributions. Note that for high NA the effects of reflection grow in importance—a fact that may explain the behavior of the $y$ component (and also the $z$ component for more extreme situations that we did not consider here).

Table 2.

Output Data for a Uniform Circular Beam of Diameter ${d}_{\mathrm{beam}}=0.375\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$, Embedded on a Quadratic Area with Length of ${d}_{x}=0.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$^{a}

Polarized solely in $x$ direction (the very small $z$ component properly calculated as well), having a wavelength of 500 nm passing an aspherical surface to a medium with ${n}_{2}=1.8$. The focal length of the asphere is therefore $f=3.375\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$. There are variations in the results that also depend on the spatial magnification factor of the image and resulting interpolation effects implicitly generated by the inverse Fourier transformation.

Values for the paraxial lens are chosen in such a way that a circular Gaussian distribution at the object will be transformed into an elliptical one and to make the rays collimated again after the paraxial element, where only the $y$ direction is modified (for a case where the user likes to propagate an elliptical profile). The quantities $d$, $A$, and $B$ should therefore obey the law of paraxial optics to stretch the beam in the $y$ direction by a demanded value. For the sake of simplicity we also did the following. To get the normalized coordinate=coordinate itself: Set the entrance pupil diameter equal to 2. To get Gaussian distribution with${\mathrm{\Delta}}_{x}=0.25$: choose Gaussian apodization in the general menu and an apodization factor of $\mathrm{G}=16$. To cut off edge rays: surface 3 properties → elliptical aperture.