Abstract

The three-dimensional frequency transfer function for optical imaging systems was introduced by Frieden in the 1960s. The analysis of this function and its partly back-transformed functions (two-dimensional and one-dimensional optical transfer functions) in the case of an ideal or aberrated imaging system has received relatively little attention in the literature. Regarding ideal imaging systems with an incoherently illuminated object volume, we present analytic expressions for the classical two-dimensional xy-transfer function in a defocused plane, for the axial z-transfer function in the presence of defocusing and for the xz-transfer function in the presence of a lateral shift δy with respect to the imaged pattern in the xz-plane. For an aberrated imaging system we use the common expansion of the aberrated pupil function with the aid of Zernike polynomials. It is shown that the line integral appearing in Frieden’s three-dimensional transfer function can be evaluated for aberrated systems using a relationship established first by Cormack between the line integral of a Zernike polynomial over a full chord of the unit disk and a Chebyshev polynomial of the second kind. Some new developments in the theory of Zernike polynomials from the last decade allow us to present explicit expressions for the line integral in the case of a weakly aberrated imaging system. We outline a similar, but more complicated, analytic scheme for the case of severely aberrated systems.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Calculation of vectorial three-dimensional transfer functions in large-angle focusing systems

Andreas Schönle and Stefan W. Hell
J. Opt. Soc. Am. A 19(10) 2121-2126 (2002)

Three-dimensional transfer functions for high-aperture systems

C. J. R. Sheppard, Min Gu, Y. Kawata, and S. Kawata
J. Opt. Soc. Am. A 11(2) 593-598 (1994)

Double Zernike expansion of the optical aberration function from its power series expansion

Joseph J. M. Braat and Augustus J. E. M. Janssen
J. Opt. Soc. Am. A 30(6) 1213-1222 (2013)

References

You do not have subscription access to this journal. Citation lists with outbound citation links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Cited By

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Figures (9)

You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Equations (58)

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription

Metrics

You do not have subscription access to this journal. Article level metrics are available to subscribers only. You may subscribe either as an OSA member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access OSA Member Subscription