Huygens–Fresnel principle and Abbe formula

Volodymyr Borovytsky

doi:10.1117/1.OE.57.9.095104

24 September 2018 Huygens–Fresnel principle and Abbe formula

Volodymyr Borovytsky

Author Affiliations +

Optical Engineering, Vol. 57, Issue 9, 095104 (September 2018). https://doi.org/10.1117/1.OE.57.9.095104

Abstract

The paper describes the approach for identification of the spatial bandwidth of an optical system in the lateral and axial directions. This approach considers application of Huygens–Fresnel principle to obtain the integral for calculation of amplitude distribution near a focal point. The replacement of integration variables leads to identification of the limits of integration in a space of spatial frequencies. These limits are the spatial cutoff frequencies for amplitude distribution. Doubling these values produces the spatial cutoff frequencies for intensity distribution, and it shows the same result as Abbe theory predicts. The proposed approach can be used for mathematical explanation why optical systems have limited spatial resolution and why the spatial harmonics with high frequencies cannot pass through optical systems. The analog of Abbe theory for axial direction is proposed. To identify the spatial bandwidth in axial direction, it has to consider the grating located in the plane with an optical axis and with slits perpendicular to this axis. Then it is possible to follow Abbe theory: to consider the diffraction on this grating, formation of a spatial spectrum, and a grating image with axial orientation in an image space.

Citation Download Citation

Volodymyr Borovytsky "Huygens–Fresnel principle and Abbe formula," Optical Engineering 57(9), 095104 (24 September 2018). https://doi.org/10.1117/1.OE.57.9.095104

Received: 24 April 2018; Accepted: 5 September 2018; Published: 24 September 2018

ACCESS THE FULL ARTICLE

INSTITUTIONAL
Select your institution to access the SPIE Digital Library.

SELECT YOUR INSTITUTION

PERSONAL
Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.

PERSONAL SIGN IN

No SPIE Account? Create one

PURCHASE THIS CONTENT

SUBSCRIBE TO DIGITAL LIBRARY

50 downloads per 1-year subscription

Members: $195

Non-members: $335 ADD TO CART

25 downloads per 1 - year subscription

Members: $145

Non-members: $250 ADD TO CART

PURCHASE SINGLE ARTICLE

Includes PDF, HTML & Video, when available