Limits...
Paraxial light distribution in the focal region of a lens: a comparison of several analytical solutions and a numerical result.

Wu Y, Kelly DP - J Mod Opt (2014)

Bottom Line: Today, it remains of significant theoretical importance for understanding the properties of imaging systems.In this manuscript, we compare and contrast each of these different solutions to a numerically calculated result, paying particular attention to how quickly each solution converges for a range of different spatial locations behind the focusing lens.These circular replicas are shown to be fundamentally different from the replicas that arise in a Cartesian coordinate system.

View Article: PubMed Central - PubMed

Affiliation: Institut für Mikro-und Nanotechnologien, Technische Universität Ilmenau , Ilmenau , Germany .

ABSTRACT

The distribution of the complex field in the focal region of a lens is a classical optical diffraction problem. Today, it remains of significant theoretical importance for understanding the properties of imaging systems. In the paraxial regime, it is possible to find analytical solutions in the neighborhood of the focus, when a plane wave is incident on a focusing lens whose finite extent is limited by a circular aperture. For example, in Born and Wolf's treatment of this problem, two different, but mathematically equivalent analytical solutions, are presented that describe the 3D field distribution using infinite sums of [Formula: see text] and [Formula: see text] type Lommel functions. An alternative solution expresses the distribution in terms of Zernike polynomials, and was presented by Nijboer in 1947. More recently, Cao derived an alternative analytical solution by expanding the Fresnel kernel using a Taylor series expansion. In practical calculations, however, only a finite number of terms from these infinite series expansions is actually used to calculate the distribution in the focal region. In this manuscript, we compare and contrast each of these different solutions to a numerically calculated result, paying particular attention to how quickly each solution converges for a range of different spatial locations behind the focusing lens. We also examine the time taken to calculate each of the analytical solutions. The numerical solution is calculated in a polar coordinate system and is semi-analytic. The integration over the angle is solved analytically, while the radial coordinate is sampled with a sampling interval of [Formula: see text] and then numerically integrated. This produces an infinite set of replicas in the diffraction plane, that are located in circular rings centered at the optical axis and each with radii given by [Formula: see text], where [Formula: see text] is the replica order. These circular replicas are shown to be fundamentally different from the replicas that arise in a Cartesian coordinate system.

No MeSH data available.


Related in: MedlinePlus

Error map of solution ,  is plotted. (The colour version of this figure is included in the online version of the journal.)
© Copyright Policy - open-access
Related In: Results  -  Collection


getmorefigures.php?uid=PMC4311934&req=5

Figure 0012: Error map of solution , is plotted. (The colour version of this figure is included in the online version of the journal.)


Paraxial light distribution in the focal region of a lens: a comparison of several analytical solutions and a numerical result.

Wu Y, Kelly DP - J Mod Opt (2014)

Error map of solution ,  is plotted. (The colour version of this figure is included in the online version of the journal.)
© Copyright Policy - open-access
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4311934&req=5

Figure 0012: Error map of solution , is plotted. (The colour version of this figure is included in the online version of the journal.)
Bottom Line: Today, it remains of significant theoretical importance for understanding the properties of imaging systems.In this manuscript, we compare and contrast each of these different solutions to a numerically calculated result, paying particular attention to how quickly each solution converges for a range of different spatial locations behind the focusing lens.These circular replicas are shown to be fundamentally different from the replicas that arise in a Cartesian coordinate system.

View Article: PubMed Central - PubMed

Affiliation: Institut für Mikro-und Nanotechnologien, Technische Universität Ilmenau , Ilmenau , Germany .

ABSTRACT

The distribution of the complex field in the focal region of a lens is a classical optical diffraction problem. Today, it remains of significant theoretical importance for understanding the properties of imaging systems. In the paraxial regime, it is possible to find analytical solutions in the neighborhood of the focus, when a plane wave is incident on a focusing lens whose finite extent is limited by a circular aperture. For example, in Born and Wolf's treatment of this problem, two different, but mathematically equivalent analytical solutions, are presented that describe the 3D field distribution using infinite sums of [Formula: see text] and [Formula: see text] type Lommel functions. An alternative solution expresses the distribution in terms of Zernike polynomials, and was presented by Nijboer in 1947. More recently, Cao derived an alternative analytical solution by expanding the Fresnel kernel using a Taylor series expansion. In practical calculations, however, only a finite number of terms from these infinite series expansions is actually used to calculate the distribution in the focal region. In this manuscript, we compare and contrast each of these different solutions to a numerically calculated result, paying particular attention to how quickly each solution converges for a range of different spatial locations behind the focusing lens. We also examine the time taken to calculate each of the analytical solutions. The numerical solution is calculated in a polar coordinate system and is semi-analytic. The integration over the angle is solved analytically, while the radial coordinate is sampled with a sampling interval of [Formula: see text] and then numerically integrated. This produces an infinite set of replicas in the diffraction plane, that are located in circular rings centered at the optical axis and each with radii given by [Formula: see text], where [Formula: see text] is the replica order. These circular replicas are shown to be fundamentally different from the replicas that arise in a Cartesian coordinate system.

No MeSH data available.


Related in: MedlinePlus